Suppose the departure angle in Fig. 3.26 is and the

A squirrel has x- and y-coordinates at time and coordinates at time For this time interval, find (a) the components of the average velocity, and (b) the magnitude and direction of the average velocity.

A rhinoceros is at the origin of coordinates at time For the time interval from to the rhinos average velocity has x-component and y-component At time (a) what are the x- and y-coordinates of the rhino? (b) How far is the rhino from the origin?

CALC A web page designer creates an animation in which a dot on a computer screen has a position of (a) Find the magnitude and direction of the dots average velocity between and (b) Find the magnitude and direction of the instantaneous velocity at and (c) Sketch the dots trajectory from to and show the velocities calculated in part (b).

CALC The position of a squirrel running in a park is given by . (a) What are and , the x- and y-components of the velocity of the squirrel, as functions of time? (b) At , how far is the squirrel from its initial position? (c) At what are the magnitude and direction of the squirrels velocity?

A jet plane is flying at a constant altitude. At time it has components of velocity At time the components are (a) Sketch the velocity vectors at and How do these two vectors differ? For this time interval calculate (b) the components of the average acceleration, and (c) the magnitude and direction of the average acceleration.

A dog running in an open field has components of velocity and at For the time interval from to the average acceleration of the dog has magnitude and direction measured from the toward the At (a) what are the x- and y-components of the dogs velocity? (b) What are the magnitude and direction of the dogs velocity? (c) Sketch the velocity vectors at and How do these two vectors differ?

The coordinates of a bird flying in the xy-plane are given by \(x(t)=\alpha t\) and \(y(t)=3.0\mathrm{\ m}-\beta t^2\) where \(\alpha=2.4\mathrm{\ m}/\mathrm{s}\) and \(\beta=1.2\mathrm{\ m}/\mathrm{s}^2\). Sketch the path of the bird between t = 0 and t = 2.0 s. (b) Calculate the velocity and acceleration vectors of the bird as functions of time. (c) Calculate the magnitude and direction of the bird's velocity and acceleration at t = 2.0 s. (d) Sketch the velocity and acceleration vectors at t = 2.0 s. At this instant, is the bird speeding up, is it slowing down, or is its speed instantaneously not changing? Is the bird turning? If so, in what direction?

CALC A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by (a) What are and , the x- and y-components of the velocity of the car as functions of time? (b) What are the magnitude and direction of the velocity of the car at ? (b) What are the magnitude and direction of the acceleration of the car at ?

A physics book slides off a horizontal tabletop with a speed of 1.10 m/s. It strikes the floor in 0.350 s. Ignore air resistance. Find (a) the height of the tabletop above the floor; (b) the horizontal distance from the edge of the table to the point where the book strikes the floor; (c) the horizontal and vertical components of the book’s velocity, and the magnitude and direction of its velocity, just before the book reaches the floor. (d) Draw \(x-t,\ y-t,\ v_x-t,\text{ and } v_{y}-t\) graphs for the motion. Text Transcription: x-t, y-t, v_x-t, and v_y-t

A daring 510-N swimmer dives off a cliff with a running horizontal leap, as shown in Fig. E3.10. What must her minimum speed be just as she leaves the top of the cliff so that she will miss the ledge at the bottom, which is 1.75 m wide and 9.00 m below the top of the cliff?

Two crickets, Chirpy and Milada, jump from the top of a vertical cliff. Chirpy just drops and reaches the ground in 3.50 s, while Milada jumps horizontally with an initial speed of How far from the base of the cliff will Milada hit the ground?

A rookie quarterback throws a football with an initial upward velocity component of and a horizontal velocity component of Ignore air resistance. (a) How much time is required for the football to reach the highest point of the trajectory? (b) How high is this point? (c) How much time (after it is thrown) is required for the football to return to its original level? How does this compare with the time calculated in part (a)? (d) How far has the football traveled horizontally during this time? (e) Draw x-t, y-t, and graphs for the motion.

Leaping the River I. A car traveling on a level horizontal road comes to a bridge during a storm and finds the bridge washed out. The driver must get to the other side, so he decides to try leaping it with his car. The side of the road the car is on is 21.3 m above the river, while the opposite side is a mere 1.8 m above the river. The river itself is a raging torrent 61.0 m wide. (a) How fast should the car be traveling at the time it leaves the road in order just to clear the river and land safely on the opposite side? (b) What is the speed of the car just before it lands on the other side?

BIO The Champion Jumper of the Insect World. The froghopper, Philaenus spumarius, holds the world record for insect jumps. When leaping at an angle of 58.0 above the horizontal, some of the tiny critters have reached a maximum height of 58.7 cm above the level ground. (See Nature, Vol. 424, July 31, 2003, p. 509.) (a) What was the takeoff speed for such a leap? (b) What horizontal distance did the froghopper cover for this world-record leap?

Inside a starship at rest on the earth, a ball rolls off the top of a horizontal table and lands a distance D from the foot of the table. This starship now lands on the unexplored Planet X. The commander, Captain Curious, rolls the same ball off the same table with the same initial speed as on earth and finds that it lands a distance 2.76D from the foot of the table. What is the acceleration due to gravity on Planet X?

On level ground a shell is fired with an initial velocity of at 60.0 above the horizontal and feels no appreciable air resistance. (a) Find the horizontal and vertical components of the shells initial velocity. (b) How long does it take the shell to reach its highest point? (c) Find its maximum height above the ground. (d) How far from its firing point does the shell land? (e) At its highest point, find the horizontal and vertical components of its acceleration and velocity.

A major leaguer hits a baseball so that it leaves the bat at a speed of 30.0 m/s and at an angle of \(36.9^{\circ}\) above the horizontal. You can ignore air resistance. (a) At what two times is the baseball at a height of 10.0 m above the point at which it left the bat? (b) Calculate the horizontal and vertical components of the baseball’s velocity at each of the two times calculated in part (a). (c) What are the magnitude and direction of the baseball’s velocity when it returns to the level at which it left the bat?

A shot putter releases the shot some distance above the level ground with a velocity of above the horizontal. The shot hits the ground 2.08 s later. You can ignore air resistance. (a) What are the components of the shots acceleration while in flight? (b) What are the components of the shots velocity at the beginning and at the end of its trajectory? (c) How far did she throw the shot horizontally? (d) Why does the expression for R in Example 3.8 not give the correct answer for part (c)? (e) How high was the shot above the ground when she released it? (f) Draw x-t, y-t, and graphs for the motion.

Win the Prize. In a carnival booth, you win a stuffed giraffe if you toss a quarter into a small dish. The dish is on a shelf above the point where the quarter leaves your hand and is a horizontal distance of 2.1 m from this point (Fig. E3.19). If you toss the coin with a velocity of 6.4 m/s at an angle of \(60^{\circ}\) above the horizontal, the coin lands in the dish. You can ignore air resistance. (a) What is the height of the shelf above the point where the quarter leaves your hand? (b) What is the vertical component of the velocity of the quarter just before it lands in the dish?

Suppose the departure angle \(\alpha_{0}\) in Fig. 3.26 is \(42.0^{\circ}\) and the distance d is 3.00 m. Where will the dart and monkey meet if the initial speed of the dart is (a) 12.0 m/s? (b) 8.0 m/s? (c) What will happen if the initial speed of the dart is 4.0 m/s? Sketch the trajectory in each case.

A man stands on the roof of a 15.0-m-tall building and throws a rock with a velocity of magnitude at an angle of above the horizontal. You can ignore air resistance. Calculate (a) the maximum height above the roof reached by the rock; (b) the magnitude of the velocity of the rock just before it strikes the ground; and (c) the horizontal range from the base of the building to the point where the rock strikes the ground. (d) Draw x-t, y-t, and graphs for the motion.

Firemen are shooting a stream of water at a burning building using a high-pressure hose that shoots out the water with a speed of as it leaves the end of the hose. Once it leaves the hose, the water moves in projectile motion. The firemen adjust the angle of elevation of the hose until the water takes 3.00 s to reach a building 45.0 m away. You can ignore air resistance; assume that the end of the hose is at ground level. (a) Find the angle of elevation (b) Find the speed and acceleration of the water at the highest point in its trajectory. (c) How high above the ground does the water strike the building, and how fast is it moving just before it hits the building?

A 124-kg balloon carrying a 22-kg basket is descending with a constant downward velocity of A 1.0-kg stone is thrown from the basket with an initial velocity of perpendicular to the path of the descending balloon, as measured relative to a person at rest in the basket. The person in the basket sees the stone hit the ground 6.00 s after being thrown. Assume that the balloon continues its downward descent with the same constant speed of (a) How high was the balloon when the rock was thrown out? (b) How high is the balloon when the rock hits the ground? (c) At the instant the rock hits the ground, how far is it from the basket? (d) Just before the rock hits the ground, find its horizontal and vertical velocity components as measured by an observer (i) at rest in the basket and (ii) at rest on the ground.

BIO Dizziness. Our balance is maintained, at least in part, by the endolymph fluid in the inner ear. Spinning displaces this fluid, causing dizziness. Suppose a dancer (or skater) is spinning at a very fast 3.0 revolutions per second about a vertical axis through the center of his head. Although the distance varies from person to person, the inner ear is approximately 7.0 cm from the axis of spin. What is the radial acceleration (in and in gs) of the endolymph fluid?

The earth has a radius of 6380 km and turns around once on its axis in 24 h. (a) What is the radial acceleration of an object at the earths equator? Give your answer in and as a fraction of g. (b) If at the equator is greater than g, objects will fly off the earths surface and into space. (We will see the reason for this in Chapter 5.) What would the period of the earths rotation have to be for this to occur?

A model of a helicopter rotor has four blades, each 3.40 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at (a) What is the linear speed of the blade tip, in (b) What is the radial acceleration of the blade tip expressed as a multiple of the acceleration of gravity, g?

BIO Pilot Blackout in a Power Dive. A jet plane comes in for a downward dive as shown in Fig. E3.27. The bottom part of the path is a quarter circle with a radius of curvature of 350 m. According to medical tests, pilots lose consciousness at an acceleration of 5.5g. At what speed (in and in mph) will the pilot black out for this dive?

The radius of the earth’s orbit around the sun (assumed to be circular) is \(1.50\times 10^8\mathrm{\ km}\) and the earth travels around this orbit in 365 days. (a) What is the magnitude of the orbital velocity of the earth, in m/s? (b) What is the radial acceleration of the earth toward the sun, in \(\mathrm{m} / \mathrm{s}^{2} \text { ? }\) (c) Repeat parts (a) and (b) for the motion of the planet Mercury (\(\text{ orbit radius }=5.79\times 10^7\mathrm{\ km}\text{, }\text{orbital period }=88.0\text{ days }\))..

A Ferris wheel with radius 14.0 m is turning about a horizontal axis through its center (Fig. E3.29). The linear speed of a passenger on the rim is constant and equal to What are the magnitude and direction of the passengers acceleration as she passes through (a) the lowest point in her circular motion? (b) The highest point in her circular motion? (c) How much time does it take the Ferris wheel to make one revolution?

BIO Hypergravity. At its Ames Research Center, NASA uses its large 20-G centrifuge to test the effects of very large accelerations (hypergravity) on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and the astronaut is strapped in at the other end. Suppose that he is aligned along the arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this machine is typically 12.5g. (a) How fast must the astronauts head be moving to experience this maximum acceleration? (b) What is the difference between the acceleration of his head and feet if the astronaut is 2.00 m tall? (c) How fast in rpm is the arm turning to produce the maximum sustained acceleration?

A moving sidewalk in an airport terminal building moves at and is 35.0 m long. If a woman steps on at one end and walks at relative to the moving sidewalk, how much time does she require to reach the opposite end if she walks (a) in the same direction the sidewalk is moving? (b) In the opposite direction?

A railroad flatcar is traveling to the right at a speed of relative to an observer standing on the ground. Someone is riding a motor scooter on the flatcar (Fig. E3.32). What is the velocity (magnitude and direction) of the motor scooter relative to the flatcar if its velocity relative to the observer on the ground is (a) to the right? (b) to the left? (c) zero?

A canoe has a velocity of southeast relative to the earth. The canoe is on a river that is flowing east relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

Two piers, A and B, are located on a river: B is 1500 m downstream from A (Fig. E3.34). Two friends must make round trips from pier A to pier B and return. One rows a boat at a constant speed of 4.00 km/h relative to the water; the other walks on the shore at a constant speed of 4.00 km/h. The velocity of the river is 2.80 km/h in the direction from A to B. How much time does it take each person to make the round trip?

Crossing the River I. A river flows due south with a speed of 2.0 m/s. A man steers a motorboat across the river; his velocity relative to the water is 4.2 m/s due east. The river is 800 m wide. (a) What is his velocity (magnitude and direction) relative to the earth? (b) How much time is required to cross the river? (c) How far south of his starting point will he reach the opposite bank?

Crossing the River II. (a) In which direction should the motorboat in Exercise 3.35 head in order to reach a point on the opposite bank directly east from the starting point? (The boats speed relative to the water remains ) (b) What is the velocity of the boat relative to the earth? (c) How much time is required to cross the river?

The nose of an ultralight plane is pointed south, and its airspeed indicator shows 35 m/s. The plane is in a 10-m/s wind blowing toward the southwest relative to the earth. (a) In a vector-addition diagram, show the relationship of \(\vec{\boldsymbol{v}}_{\mathrm{P} / \mathrm{E}}\) (the velocity of the plane relative to the earth) to the two given vectors. (b) Letting x be east and y be north, find the components of \(\vec{\boldsymbol{v}}_{\mathrm{P} / \mathrm{E}}\). (c) Find the magnitude and direction of \(\vec{\boldsymbol{v}}_{\mathrm{P} / \mathrm{E}}\).

An airplane pilot wishes to fly due west. A wind of 80.0 km/h (about 50 mi/h ) is blowing toward the south. (a) If the airspeed of the plane (its speed in still air) is (about 200 mi/h), in which direction should the pilot head? (b) What is the speed of the plane over the ground? Illustrate with a vector diagram.

Bird Migration. Canadian geese migrate essentially along a north-south direction for well over a thousand kilometers in some cases, traveling at speeds up to about 100 km/h. If one such bird is flying at 100 km/h relative to the air, but there is a 40 km/h wind blowing from west to east, (a) at what angle relative to the north-south direction should this bird head so that it will be traveling directly southward relative to the ground? (b) How long will it take the bird to cover a ground distance of 500 km from north to south? (Note: Even on cloudy nights, many birds can navigate using the earth's magnetic field to fix the north-south direction.)

An athlete starts at point A and runs at a constant speed of around a circular track 100 m in diameter, as shown in Fig. P3.40. Find the xand y-components of this runners average velocity and average acceleration between points (a) A and B, (b) A and C, (c) C and D, and (d) A and A (a full lap). (e) Calculate the magnitude of the runners average velocity between A and B. Is his average speed equal to the magnitude of his average velocity? Why or why not? (f) How can his velocity be changing if he is running at constant speed?

CALC A rocket is fired at an angle from the top of a tower of height Because of the design of the engines, its position coordinates are of the form and where A, B, C, and D are constants. Furthermore, the acceleration of the rocket 1.00 s after firing is Take the origin of coordinates to be at the base of the tower. (a) Find the constants A, B, C, and D, including their SI units. (b) At the instant after the rocket is fired, what are its acceleration vector and its velocity? (c) What are the x- and y-components of the rockets velocity 10.0 s after it is fired, and how fast is it moving? (d) What is the position vector of the rocket 10.0 s after it is fired?

CALC A faulty model rocket moves in the xy-plane (the positive y-direction is vertically upward). The rockets acceleration has components and where and At the rocket is at the origin and has velocity with and (a) Calculate the velocity and position vectors as functions of time. (b) What is the maximum height reached by the rocket? (c) Sketch the path of the rocket. (d) What is the horizontal displacement of the rocket when it returns to

CALC If \(\vec{r}=bt^2\hat{i}+ct^3\hat{j}\) where b and c are positive constants, when does the velocity vector make an angle of 45.0° with the x - and y -axes?

A small toy airplane is flying in the xy-plane parallel to the ground. In the time interval to , its velocity as a function of time is given by . At what value of Is the velocity of the plane perpendicular to its acceleration?

. CP CALC A small toy airplane is flying in the xy-plane parallel to the ground. In the time interval to , its velocity as a function of time is given by v . At what value of t is the velocity of the plane perpendicular to its acceleration?

A bird flies in the xy-plane with a velocity vector given by \(\vec{\boldsymbol{v}}=\left(\alpha-\beta t^{2}\right) \hat{\boldsymbol{\imath}}+\gamma t \hat{\boldsymbol{j}}\), with \(\alpha=2.4\mathrm{\ m}/\mathrm{s},\ \beta=1.6\mathrm{\ m}/\mathrm{s}^3\), and \(\gamma=4.0\mathrm{\ m}/\mathrm{s}^2\). The positive y-direction is vertically upward. At t = 0 the bird is at the origin. (a) Calculate the position and acceleration vectors of the bird as functions of time. (b) What is the bird’s altitude (y-coordinate) as it flies over x = 0 for the first time after t = 0?

A test rocket is launched by accelerating it along a 200.0-m incline at starting from rest at point A (Fig. P3.47). The incline rises at 35.0 above the horizontal, and at the instant the rocket leaves it, its engines turn off and it is subject only to gravity (air resistance can be ignored). Find (a) the maximum height above the ground that the rocket reaches, and (b) the greatest horizontal range of the rocket beyond point A.

In the long jump, an athlete launches herself at an angle above the ground and lands at the same height, trying to travel the greatest horizontal distance. Suppose that on earth she is in the air for time T, reaches a maximum height h, and achieves a horizontal distance D. If she jumped in exactly the same way during a competition on Mars. where \(g_{Mars}\) is 0.379 or its earth value. find her time in the air, maximum height, and horizontal distance. Express each of these three quantities in terms or its earth value. Air resistance can be neglected on both planets.

Dynamite! A demolition crew uses dynamite to blow an old building apart. Debris from the explosion flies off in all directions and is later found at distances as far as 50 m from the explosion. Estimate the maximum speed at which debris was blown outward by the explosion. Describe any assumptions that you make.

BIO Spiraling Up. It is common to see birds of prey rising upward on thermals. The paths they take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume a bird completes a circle of radius 6.00 m every 5.00 s and rises vertically at a constant rate of Determine: (a) the speed of the bird relative to the ground; (b) the birds acceleration (magnitude and direction); and (c) the angle between the birds velocity vector and the horizontal.

A jungle veterinarian with a blow-gun loaded with a tranquilizer dart and a sly 1.5-kg monkey are each 25 m above the ground in trees 70 m apart. Just as the hunter shoots horizontally at the monkey, the monkey drops from the tree in a vain attempt to escape being hit. What must the minimum muzzle velocity of the dart have been for the hunter to have hit the monkey before it reached the ground?

A movie stuntwoman drops from a helicopter that is 30.0 m above the ground and moving with a constant velocity whose components are 10.0 m/s upward and 15.0 m/s horizontal and toward the south. You can ignore air resistance. (a) Where on the ground (relative to the position of the helicopter when she drops) should the stuntwoman have placed the foam mats that break her fall? (b) Draw x-t, y-t, \(v_{x}-t\), and \(v_{y}-t\) graphs of her motion. Text Transcription: v_x-t v_y-t

In fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. A pilot is practicing In fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. A pilot is practicing

A cannon, located 60.0 m from the base of a vertical 25.0-m-tall cliff, shoots a 15-kg shell at \(43.0^{\circ}\) above the horizontal toward the cliff. (a) What must the minimum muzzle velocity be for the shell to clear the top of the cliff? (b) The ground at the top of the cliff is level, with a constant elevation of 25.0 m above the cannon. Under the conditions of part (a), how far does the shell land past the edge of the cliff?

An airplane is flying with a velocity of 90.0 m/s at an angle of \(23.0^{\circ}\) above the horizontal. When the plane is 114 m directly above a dog that is standing on level ground, a suitcase drops out of the luggage compartment. How far from the dog will the suitcase land? You can ignore air resistance.

As a ship is approaching the dock at 45.0 cm/s, an important piece of landing equipment needs to be thrown to it before it can dock. This equipment is thrown at 15.0 m/s at \(60.0^{\circ}\) above the horizontal from the top of a tower at the edge of the water, 8.75 m above the ship's deck (Fig. P3.56). For this equipment to land at the front of the ship, at what distance D from the dock should the ship be when the equipment is thrown? Air resistance can be neglected.

A toy rocket is launched with an initial velocity of 12.0 m s in the horizontal direction from the roof of a 30.0-m-tall building. The rockets engine produces a horizontal acceleration of , in the same direction as the initial velocity, but in the vertical direction the acceleration is g, downward. Air resistance can be neglected. What horizontal distance does the rocket travel before reaching the ground?

An Errand of Mercy. An airplane is dropping bales of hay to cattle stranded in a blizzard on the Great Plains. The pilot releases the bales at 150 m above the level ground when the plane is flying at in a direction 55 above the horizontal. How far in front of the cattle should the pilot release the hay so that the bales land at the point where the cattle are stranded?

The Longest Home Run. According to the Guinness Book of World Records, the longest home run ever measured was hit by Roy Dizzy Carlyle in a minor league game. The ball traveled 188 m (618 ft) before landing on the ground outside the ballpark. (a) Assuming the balls initial velocity was in a direction above the horizontal and ignoring air resistance, what did the initial speed of the ball need to be to produce such a home run if the ball was hit at a point 0.9 m (3.0 ft) above ground level? Assume that the ground was perfectly flat. (b) How far would the ball be above a fence 3.0 m (10 ft) high if the fence was 116 m (380 ft) from home plate?

A water hose is used to fill a large cylindrical storage tank of diameter D and height 2D. The hose shoots the water at 45 above the horizontal from the same level as the base of the tank and is a distance 6D away (Fig. P3.60). For what range of launch speeds will the water enter the tank? Ignore air resistance, and express your answer in terms of D and g.

A projectile is being launched from ground level with no air resistance. You want to avoid having it enter a temperature inversion layer in the atmosphere a height h above the ground. (a) What is the maximum launch speed you could give this projectile if you shot it straight up? Express your answer in terms of h and g. (b) Suppose the launcher available shoots projectiles at twice the maximum launch speed you found in part (a). At what maximum angle above the horizontal should you launch the projectile? (c) How far (in terms of h) from the launcher does the projectile in part (b) land?

Kicking a Field Goal. In U.S. football, after a touchdown the team has the opportunity to earn one more point by kicking the ball over the bar between the goal posts. The bar is 10.0 ft above the ground, and the ball is kicked from ground level, 36.0 ft horizontally from the bar (Fig. P3.62). Football regulations are stated in English units, but convert them to SI units for this problem. (a) There is a minimum angle above the ground such that if the ball is launched below this angle, it can never clear the bar, no matter how fast it is kicked. What is this angle? (b) If the ball is kicked at 45.0 above the horizontal, what must its initial speed be if it is to just clear the bar? Express your answer in and in km m>s >h.

A grasshopper leaps into the air from the edge of a vertical cliff, as shown in Fig. P3.63. Use information from the figure to find (a) the initial speed of the grasshopper and (b) the height of the cliff.

A World Record. In the shot put, a standard trackand-field event, a 7.3-kg object (the shot) is thrown by releasing it at approximately 40 over a straight left leg. The world record for distance, set by Randy Barnes in 1990, is 23.11 m. Assuming that Barnes released the shot put at 40.0 from a height of 2.00 m above the ground, with what speed, in and in mph, did he release it?

Look Out! A snowball rolls off a barn roof that slopes downward at an angle of (Fig. P3.65). The edge of the roof is 14.0 m above the ground, and the snowball has a speed of as it rolls off the roof. Ignore air resistance. (a) How far from the edge of the barn does the snowball strike the ground if it doesnt strike anything else while falling? (b) Draw x-t, y-t, and graphs for the motion in part (a). (c) A man 1.9 m tall is standing 4.0 m from the edge of the barn. Will he be hit by the snowball?

On the Flying Trapeze. A new circus act is called the Texas Tumblers. Lovely Mary Belle swings from a trapeze, projects herself at an angle of and is supposed to be caught by Joe Bob, whose hands are 6.1 m above and 8.2 m horizontally from her launch point (Fig. P3.66). You can ignore air resistance. (a) What initial speed must Mary Belle have just to reach Joe Bob? (b) For the initial speed calculated in part (a), what are the magnitude and direction of her velocity when Mary Belle reaches Joe Bob? (c) Assuming that Mary Belle has the initial speed calculated in part (a), draw x-t, y-t, and graphs showing the motion of both tumblers. Your graphs should show the motion up until the point where Mary Belle reaches Joe Bob. (d) The night of their debut performance, Joe Bob misses her completely as she flies past. How far horizontally does Mary Belle travel, from her initial launch point, before landing in the safety net 8.6 m below her starting point?

Leaping the River II. A physics professor did daredevil stunts in his spare time. His last stunt was an attempt to jump across a river on a motorcycle (Fig. P3.67). The takeoff ramp was inclined at the river was 40.0 m wide, and the far bank was 15.0 m lower than the top of the ramp. The river itself was 100 m below the ramp. You can ignore air resistance. (a) What should his speed have been at the top of the ramp to have just made it to the edge of the far bank? (b) If his speed was only half the value found in part (a), where did he land?

A rock is thrown from the roof of a building with a velocity \(\nu_0\) at an angle of \(\alpha_0\) from the horizontal. The building has height h. You can ignore air resistance. Calculate the magnitude of the velocity of the rock just before it strikes the ground, and show that this speed is independent of \(\alpha_0\).

A 5500-kg cart carrying a vertical rocket launcher moves to the right at a constant speed of along a horizontal track. It launches a 45.0-kg rocket vertically upward with an initial speed of relative to the cart. (a) How high will the rocket go? (b) Where, relative to the cart, will the rocket land? (c) How far does the cart move while the rocket is in the air? (d) At what angle, relative to the horizontal, is the rocket traveling just as it leaves the cart, as measured by an observer at rest on the ground? (e) Sketch the rockets trajectory as seen by an observer (i) stationary on the cart and (ii) stationary on the ground.

A 2.7-kg ball is thrown upward with an initial speed of 20.0 m/s from the edge of a 45.0-m-high cliff. At the instant the ball is thrown, a woman starts running away from the base of the cliff with a constant speed of 6.00 m/s. The woman runs in a straight line on level ground, and air resistance acting on the ball can be ignored. (a) At what angle above the horizontal should the ball be thrown so that the runner will catch it just before it hits the ground, and how far does the woman run before she catches the ball? (b) Carefully sketch the ball’s trajectory as viewed by (i) a person at rest on the ground and (ii) the runner.

A 76.0-kg boulder is rolling horizontally at the top of a vertical cliff that is 20 m above the surface of a lake, as shown in Fig. P3.71. The top of the vertical face of a dam is located 100 m from the foot of the cliff, with the top of the dam level with the surface of the water in the lake. A level plain is 25 m below the top of the dam. (a) What must be the minimum speed of the rock just as it leaves the cliff so it will travel to the plain without striking the dam? (b) How far from the foot of the dam does the rock hit the plain?

Tossing Your Lunch. Henrietta is going off to her physics class, jogging down the sidewalk at Her husband Bruce suddenly realizes that she left in such a hurry that she forgot her lunch of bagels, so he runs to the window of their apartment, which is 38.0 m above the street level and directly above the sidewalk, to throw them to her. Bruce throws them horizontally 9.00 s after Henrietta has passed below the window, and she catches them on the run. You can ignore air resistance. (a) With what initial speed must Bruce throw the bagels so Henrietta can catch them just before they hit the ground? (b) Where is Henrietta when she catches the bagels?

Two tanks are engaged in a training exercise on level ground. The first tank fires a paint-filled training round with a muzzle speed of 250 m s at above the horizontal while advancing toward the second tank with a speed of relative to the ground. The second tank is retreating at relative to the ground, but is hit by the shell. You can ignore air resistance and assume the shell hits at the same height above ground from which it was fired. Find the distance between the tanks (a) when the round was first fired and (b) at the time of impact.

Bang! A student sits atop a platform a distance h above the ground. He throws a large firecracker horizontally with a speed However, a wind blowing parallel to the ground gives the firecracker a constant horizontal acceleration with magnitude a. This results in the firecracker reaching the ground directly under the student. Determine the height h in terms of a, and g. You can ignore the effect of air resistance on the vertical motion

Bang! A student sits atop a platform a distance h above the ground. He throws a large firecracker horizontally with a speed However, a wind blowing parallel to the ground gives the firecracker a constant horizontal acceleration with magnitude a. This results in the firecracker reaching the ground directly under the student. Determine the height h in terms of a, and g. You can ignore the effect of air resistance on the vertical motion

When it is 145 m above the ground, a rocket traveling vertically upward at a constant relative to the ground launches a secondary rocket at a speed of at an angle of 53.0 above the horizontal, both quantities being measured by an astronaut sitting in the rocket. After it is launched the secondary rocket is in free-fall. (a) Just as the secondary rocket is launched, what are the horizontal and vertical components of its velocity relative to (i) the astronaut sitting in the rocket and (ii) Mission Control on the ground? (b) Find the initial speed and launch angle of the secondary rocket as measured by Mission Control. (c) What maximum height above the ground does the secondary rocket reach?

In an action-adventure film, the hero is supposed to throw a grenade from his car, which is going 90.0 km/h, to his enemy’s car, which is going 110 km/h. The enemy’s car is 15.8 m in front of the hero’s when he lets go of the grenade. If the hero throws the grenade so its initial velocity relative to him is at an angle of \(45^{\circ}\) above the horizontal, what should the magnitude of the initial velocity be? The cars are both traveling in the same direction on a level road. You can ignore air resistance. Find the magnitude of the velocity both relative to the hero and relative to the earth

A 400.0-m-wide river flows from west to east at 30.0 m/min. Your boat moves at 100.0 m/min relative to the water no matter which direction you point it. To cross this river, you start from a dock at point A on the south bank. There is a boat landing directly opposite at point B on the north bank, and also one at point C, 75.0 m downstream from B (Fig. P3.78). (a) Where on the north shore will you land if you point your boat perpendicular to the water current, and what distance will you have traveled? (b) If you initially aim your boat directly toward point C and do not change that bearing relative to the shore, where on the north shore will you land? (c) To reach point C: (i) at what bearing must you aim your boat, (ii) how long will it take to cross the river, (iii) what distance do you travel, and (iv) and what is the speed of your boat as measured by an observer standing on the river bank?

Cycloid. A particle moves in the xy-plane. Its coordinates are given as functions of time by where R and are constants. (a) Sketch the trajectory of the particle. (This is the trajectory of a point on the rim of a wheel that is rolling at a constant speed on a horizontal surface. The curve traced out by such a point as it moves through space is called a cycloid.) (b) Determine the velocity components and the acceleration components of the particle at any time t. (c) At which times is the particle momentarily at rest? What are the coordinates of the particle at these times? What are the magnitude and direction of the acceleration at these times? (d) Does the magnitude of the acceleration depend on time? Compare to uniform circular motion.

A projectile is fired from point A at an angle above the horizontal. At its highest point, after having traveled a horizontal distance D from its launch point, it suddenly explodes into two identical fragments that travel horizontally with equal but opposite velocities as measured relative to the projectile just before it exploded. If one fragment lands back at point A, how far from A (in terms of D) does the other fragment land?

An airplane pilot sets a compass course due west and maintains an airspeed of After flying for 0.500 h, she finds herself over a town 120 km west and 20 km south of her starting point. (a) Find the wind velocity (magnitude and direction). (b) If the wind velocity is due south, in what direction should the pilot set her course to travel due west? Use the same airspeed of 220 km>h.

Raindrops. When a train’s velocity is 12.0 m/s eastward, raindrops that are falling vertically with respect to the earth make traces that are inclined \(30.0^{\circ}\) to the vertical on the windows of the train. (a) What is the horizontal component of a drop’s velocity with respect to the earth? With respect to the train? (b) What is the magnitude of the velocity of the raindrop with respect to the earth? With respect to the train?

In a World Cup soccer match, Juan is running due north toward the goal with a speed of 8.00 m/s relative to the ground. A teammate passes the ball to him. The ball has a speed of 12.0 m/s and is moving in a direction \(37.0^{\circ}\) east of north, relative to the ground. What are the magnitude and direction of the ball’s velocity relative to Juan?

An elevator is moving upward at a constant speed of A bolt in the elevator ceiling 3.00 m above the elevator floor works loose and falls. (a) How long does it take for the bolt to fall to the elevator floor? What is the speed of the bolt just as it hits the elevator floor (b) according to an observer in the elevator? (c) According to an observer standing on one of the floor landings of the building? (d) According to the observer in part (c), what distance did the bolt travel between the ceiling and the floor of the elevator?

Suppose the elevator in Problem 3.84 starts from rest and maintains a constant upward acceleration of and the bolt falls out the instant the elevator begins to move. (a) How long does it take for the bolt to reach the floor of the elevator? (b) Just as it reaches the floor, how fast is the bolt moving according to an observer (i) in the elevator? (ii) Standing on the floor landings of the building? (c) According to each observer in part (b), how far has the bolt traveled between the ceiling and floor of the elevator?

Two soccer players, Mia and Alice, are running as Alice passes the ball to Mia. Mia is running due north with a speed of 6.00 m s. The velocity of the ball relative to Mia is 5.00 m/s in a direction east of south. What are the magnitude and direction of the velocity of the ball relative to the ground?

Projectile Motion on an Incline. Refer to the Bridging Problem in Chapter 3. (a) An archer on ground that has a constant upward slope of aims at a target 60.0 m farther up the incline. The arrow in the bow and the bulls-eye at the center of the target are each 1.50 m above the ground. The initial velocity of the arrow just after it leaves the bow has magnitude At what angle above the horizontal should the archer aim to hit the bullseye? If there are two such angles, calculate the smaller of the two. You might have to solve the equation for the angle by iteration that is, by trial and error. How does the angle compare to that required when the ground is level, with 0 slope? (b) Repeat the problem for ground that has a constant downward slope of 30.0.

A projectile is thrown from a point P. It moves in such a way that its distance from P is always increasing. Find the maximum angle above the horizontal with which the projectile could have been thrown. You can ignore air resistance.

Two students are canoeing on a river. While heading upstream, they accidentally drop an empty bottle overboard. They then continue paddling for 60 minutes, reaching a point 2.0 km farther upstream. At this point they realize that the bottle is missing and, driven by ecological awareness, they turn around and head downstream. They catch up with and retrieve the bottle (which has been moving along with the current) 5.0 km downstream from the turn-around point. (a) Assuming a constant paddling effort throughout, how fast is the river flowing? (b) What would the canoe speed in a still lake be for the same paddling effort?

A rocket designed to place small payloads into orbit is carried to an altitude of 12.0 km above sea level by a converted airliner. When the airliner is flying in a straight line at a constant speed of 850 km/h, the rocket is dropped. After the drop, the airliner maintains the same altitude and speed and continues to fly in a straight line. The rocket falls for a brief time, after which its rocket motor turns on. Once its rocket motor is on, the combined effects of thrust and gravity give the rocket a constant acceleration of magnitude 3.00g directed at an angle of \(30.0^{\circ}\) above the horizontal. For reasons of safety, the rocket should be at least 1.00 km in front of the airliner when it climbs through the airliner's altitude. Your job is to determine the minimum time that the rocket must fall before its engine starts. You can ignore air resistance. Your answer should include (i) a diagram showing the flight paths of both the rocket and the airliner, labeled at several points with vectors for their velocities and accelerations; (ii) an x-t graph showing the motions of both the rocket and the airliner; and (iii) a y-t graph showing the motions of both the rocket and the airliner. In the diagram and the graphs, indicate when the rocket is dropped, when the rocket motor turns on, and when the rocket climbs through the altitude of the airliner.

Problem 1DQ A simple pendulum (a mass swinging at the end of a string) swings back and forth in a circular arc. What is the direction of the acceleration of the mass when it is at the ends of the swing? At the midpoint? In each case, explain how you obtained your answer.

A squirrel has x- and y-coordinates (1.1 m, 3.4 m) at time \(t_1\) = 0 and coordinates (5.3 m, - 0.5 m) at time \(t_2\) = 3.0 s. For this time interval, find (a) the components of the average velocity, and (b) the magnitude and direction of the average velocity.

Redraw Fig. 3.11a if \(\vec{a}\) is antiparallel to \(\vec{v}_{1}\). Does the particle move in a straight line? What happens to its speed? Equation Transcription: Text Transcription: Vec a Vec v1

Problem 2E A rhinoceros is at the origin of coordinates at time t1 = 0. For the time interval from t1 = 0 to t2 = 12.0 s, the rhino’s average velocity has x-component - 3.8 m/s and y-component 4.9 m/s. At time t2 = 12.0 s, (a) what are the x- and y-coordinates of the rhino? (b) How far is the rhino from the origin?

A projectile moves in a parabolic path without air resistance. Is there any point at which \(\vec{a}\) is parallel to \(\vec{v}\)? Perpendicular to \(\vec{v}\)? Explain. Equation Transcription: Text Transcription: Vec{a} Vec{v} vec{v}

CALC A web page designer creates an animation in which a dot on a computer screen has a position of \(\vec{r}=\left[4.0 \mathrm{~cm}+\left(2.5 \mathrm{~cm} / \mathrm{s}^{2}\right) t^{2}\right] \hat{\imath}+(5.0 \mathrm{~cm} / \mathrm{s}) t \hat{\jmath}\) (a) Find the magnitude and direction of the dot’s average velocity between \(t=0 \text { and } t=2.0 \mathrm{~s} \text {. }\). (b) Find the magnitude and direction of the instantaneous velocity at \(t=0, t=1.0 \mathrm{~s}, \text { and } t=2.0 \mathrm{~s}\). (c) Sketch the dot’s trajectory from \(t=\text { to } t=2.0 \mathrm{~s}\), and show the velocities calculated in part (b). Equation Transcription: + Text Transcription: r=4.0 cm+(2.5 cm/s2) t2î+ (5.0 cm/s)t? t=0 and t=2.0 s t=0, t=1.0 s, and t=2.0 s t=tot=2.0 s

When a rifle is fired at a distant target, the barrel is not lined up exactly on the target. Why not? Does the angle of correction depend on the distance to the target?

CALC The position of a squirrel running in a park is given by \(\vec{r}=\left[(0.280 \mathrm{~m} / \mathrm{s}) t+\left(0.360 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}\right] \hat{\imath}+\left(0.0190 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3} \hat{\jmath} .\) (a) What are \(v_{x}(t) \text { and } v_{y}(t)\), the \(\chi \text { - and } y \text {-components }\)of the velocity of the squirrel, as functions of time? (b) At ???? = 5.00 s, how far is the squirrel from its initial position? (c) At ???? = 5.00 s, what are the magnitude and direction of the squirrel’s velocity? Equation Transcription: Text Transcription: r= (0.280 m/s)t +(0.360m/s2)t2î+(0.0190 m/s3)t3 ? vx(t) and vy (t) x- and y-components

Problem 5DQ At the instant that you fire a bullet horizontally from a rifle, you drop a bullet from the height of the gun barrel. If there is no air resistance, which bullet hits the level ground first? Explain.

Problem 5E A jet plane is flying at a constant altitude. At time t1 = 0, it has components of velocity vx = 90 m/s, vy = 110 m/s. At time t2 = 30.0 s, the components are vx = - 170 m/s, vy = 40 m/s. (a) Sketch the velocity vectors at t1 and t2. How do these two vectors differ? For this time interval calculate (b) the components of the average acceleration, and (c) the magnitude and direction of the average acceleration.

A package falls out of an airplane that is flying in a straight line at a constant altitude and speed. If you ignore air resistance, what would be the path of the package as observed by the pilot? As observed by a person on the ground?

Problem 6E A dog running in an open field has components of velocity vx = 2.6 m/s and vy = -1.8 m/s at t1 = 10.0 s. For the time interval from t1 = 10.0 s to t2 = 20.0 s, the average acceleration of the dog has magnitude 0.45 m/s2 and direction 31.0o measured from the + x-axis toward the + y-axis. At t2 = 20.0 s, (a) what are the x- and y-components of the dog’s velocity? (b) What are the magnitude and direction of the dog’s velocity? (c) Sketch the velocity vectors at t1 and t2. How do these two vectors differ?

Problem 7DQ Sketch the six graphs of the x- and y-components of position, velocity, and acceleration versus time for projectile motion with x0 = y0 = 0 and 0 < ?0 < 90o.

Problem 7E CALC The coordinates of a bird flying in the xy-plane are given by x(t) = ?t and y(t) = 3.0 m - ?t2, where ? = 2.4 m/s and ? = 1.2 m/s2. (a) Sketch the path of the bird between t = 0 and t = 2.0 s. (b) Calculate the velocity and acceleration vectors of the bird as functions of time. (c) Calculate the magnitude and direction of the bird’s velocity and acceleration at t = 2.0 s. (d) Sketch the velocity and acceleration vectors at t = 2.0 s. At this instant, is the bird’s speed increasing, decreasing, or not changing? Is the bird turning? If so, in what direction?

If a jumping frog can give itself the same initial speed regardless of the direction in which it jumps (forward or straight up), how is the maximum vertical height to which it can jump related to its maximum horizontal range \(R_{\max}=v_0^{\ 2}/g\)?

A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by \(\vec{v}=\left[5.00 \mathrm{\ m} / \mathrm{s}-\left(0.0180 \mathrm{\ m} / \mathrm{s}^{3}\right) t^{2}\right] \hat{\imath}+\left[2.00 \mathrm{\ m} / \mathrm{s}+\left(0.550 \mathrm{\ m} / \mathrm{s}^{2}\right) t\right] \hat{J}\). (a) What are \(a_{x}(t)\) and \(a_{y}(t)\), the x- and y-components of the velocity of the car as functions of time? (b) What are the magnitude and direction of the velocity of the car at t = 8.00 s? (b) What are the magnitude and direction of the acceleration of the car at t = 8.00 s?

Problem 9DQ A projectile is fired upward at an angle ? above the horizontal with an initial speed v0. At its maximum height, what are its velocity vector, its speed, and its acceleration vector?

A physics book slides off a horizontal tabletop with a speed of 1.10 m/s. It strikes the floor in 0.350 s. Ignore air resistance. Find (a) the height of the tabletop above the floor; (b) the horizontal at distance from the edge of the table to the point where the book strikes the floor; (c) the horizontal and vertical components of the book’s velocity, and the magnitude and direction of its velocity, just before the book reaches the floor. (d) Draw x-t, y-t, vx-t, and vy-t graphs for the motion.

Problem 10DQ In uniform circular motion, what are the average velocity and average acceleration for one revolution? Explain.

A daring 510-N swimmer dives off a cliff with a running horizontal leap, as shown in Fig. E3.10. What must her minimum speed be just as she leaves the top of the cliff so that she will miss the ledge at the bottom, which is \(1.75 \mathrm{~m}\) wide and \(9.00 \mathrm{~m}\) below the top of the cliff? Equation Transcription: Text Transcription: 1.75 m 9.00 m

Problem 11DQ In uniform circular motion, how does the acceleration change when the speed is increased by a factor of 3? When the radius is decreased by a factor of 2?

Problem 11E Two crickets, Chirpy and Milada, jump from the top of a vertical cliff. Chirpy just drops and reaches the ground in 3.50 s, while Milada jumps horizontally with an initial speed of 95.0 cm/s. How far from the base of the cliff will Milada hit the ground?

Problem 12E A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. (a) How much time is required for the football to reach the highest point of the trajectory? (b) How high is this point? (c) How much time (after it is thrown) is required for the football to return to its original level? How does this compare with the time calculated in part (a)? (d) How far has the football traveled horizontally during this time? (e) Draw x-t, y-t, vx-t, and vy-t graphs for the motion.

Problem 13DQ Raindrops hitting the side windows of a car in motion often leave diagonal streaks even if there is no wind. Why? Is the explanation the same or different for diagonal streaks on the windshield?

Leaping the River I. A car traveling on a level horizontal road comes to a bridge during a storm and finds the bridge washed out. The driver must get to the other side, so he decides to try leaping it with his car. The side of the road the car is on is 21.3 m above the river, while the opposite side is a mere 1.8 m above the river. The river itself is a raging torrent 61.0 m wide. (a) How fast should the car be traveling at the time it leaves the road in order just to clear the river and land safely on the opposite side? (b) What is the speed of the car just before it lands on the other side?

In a rainstorm with a strong wind, what determines the best position in which to hold an umbrella?

Problem 14E BIO The Champion Jumper of the Insect World. The froghopper, Philaenus spumarius, holds the world record for insect jumps. When leaping at an angle of 58.0° above the horizontal, some of the tiny critters have reached a maximum height of 58.7 cm above the level ground. (See Nature, Vol. 424, July 31, 2003, p. 509.) (a) What was the takeoff speed for such a leap? (b) What horizontal distance did the froghopper cover for this world-record leap?

Problem 15DQ You are on the west bank of a river that is flowing north with a speed of 1.2 m/s. Your swimming speed relative to the water is 1.5 m/s, and the river is 60 m wide. What is your path relative to the earth that allows you to cross the river in the shortest time? Explain your reasoning.

Inside a starship at rest on the earth, a ball rolls off the top of a horizontal table and lands a distance D from the foot of the table. This starship now lands on the unexplored Planet X. The commander, Captain Curious, rolls the same ball off the same table with the same initial speed as on earth and finds that it lands a distance 2.76D from the foot of the table. What is the acceleration due to gravity on Planet X?

A stone is thrown into the air at an angle above the horizontal and feels negligible air resistance. Which graph in Fig. Q3.16 best depicts the stone’s speed \(v\) as a function of time \(t\) while it is in the air? Equation Transcription: Text Transcription: v t

On level ground a shell is fired with an initial velocity of 50.0 m/s at \(60.0^{\circ}\) above the horizontal and feels no appreciable air resistance. (a) Find the horizontal and vertical components of the shell’s initial velocity. (b) How long does it take the shell to reach its highest point? (c) Find its maximum height above the ground. (d) How far from its firing point does the shell land? (e) At its highest point, find the horizontal and vertical components of its acceleration and velocity.

A major leaguer hits a baseball so that it leaves the bat at a speed of 30.0 m/s and at an angle of 36.9° above the horizontal. You can ignore air resistance. (a) At what two times is the baseball at a height of 10.0 m above the point at which it left the bat? (b) Calculate the horizontal and vertical components of the baseball’s velocity at each of the two times calculated in part (a). (c) What are the magnitude and direction of the baseball’s velocity when it returns to the level at which it left the bat?

A shot putter releases the shot some distance above the level ground with a velocity of \(12.0 \mathrm{~m} / \mathrm{s}, 51.0^{\circ}\)above the horizontal. The shot hits the ground 2.08 s later. You can ignore air resistance. (a) What are the components of the shot’s acceleration while in flight? (b) What are the components of the shot’s velocity at the beginning and at the end of its trajectory? (c) How far did she throw the shot horizontally? (d) Why does the expression for R in Example 3.8 not give the correct answer for part (c)? (e) How high was the shot above the ground when she released it? (f) Draw \(\chi-t, y^{-t}, v_{y} t\) graphs the motion. Equation Transcription: ° Text Transcription: 12.0 m/s, 51.0° x-x, y-t, vyt

Win the Prize. In a carnival booth, you win a stuffed giraffe if you toss a quarter into a small dish. The dish is on a shelf above the point where the quarter leaves your hand and is a horizontal distance of \(2.1 \mathrm{~m}\) from this point (Fig. E3.19). If you toss the coin with a velocity of \(6.4 \mathrm{~m} / \mathrm{s}\) at an angle of \(60^{\circ}\) above the horizontal, the coin lands in the dish. You can ignore air resistance. (a) What is the height of the shelf above the point where the quarter leaves your hand? (b) What is the vertical component of the velocity of the quarter just before it lands in the dish? Equation Transcription: ° Text Transcription: 2.1 m 6.4 m/s 60°

Suppose the departure angle \(\text { o } 0\) in Fig. 3.26 is \(42.0^{\circ}\) and the distance d is 3.00 m. Where will the dart and monkey meet if the initial speed of the dart is (a) 12.0 m/s? (b) 8.0 m/s? (c) What will happen if the initial speed of the dart is 4.0 m/s? Sketch the trajectory in each case. Equation Transcription: 42.0° Text Transcription: a0 42.0^°

Problem 21E A man stands on the roof of a 15.0-m-tall building and throws a rock with a speed of 30.0 m/s at an angle of 33.0o above the horizontal. Ignore air resistance. Calculate (a) the maximum height above the roof that the rock reaches; (b) the speed of the rock just before it strikes the ground; and (c) the horizontal range from the base of the building to the point where the rock strikes the ground. (d) Draw x-t, y-t, vx-t, and vy-t graphs for the motion.

Firemen use a high-pressure hose to shoot a stream of water at a burning building. The water has a speed of 25.0 m/s as it leaves the end of the hose and then exhibits projectile motion. The firemen adjust the angle of elevation ? of the hose until the water takes 3.00 s to reach a building 45.0 m away. Ignore air resistance; assume that the end of the hose is at ground level. (a) Find ?. (b) Find the speed and acceleration of the water at the highest point in its trajectory. (c) How high above the ground does the water strike the building, and how fast is it moving just before it hits the building?

Problem 23E A 124-kg balloon carrying a 22-kg basket is descending with a constant downward velocity of 20.0 m/s. A 1.0-kg stone is thrown from the basket with an initial velocity of 15.0 m/s perpendicular to the path of the descending balloon as measured relative to a person at rest in the basket. The person in the basket sees the stone hit the ground 6.00 s after being thrown. Assume that the balloon continues its downward descent with the same constant speed of 20.0 m/s. (a) How high was the balloon when the rock was thrown out? (b) How high is the balloon when the rock hits the ground? (c) At the instant the rock hits the ground, how far is it from the basket? (d) Just before the rock hits the ground, find its horizontal and vertical velocity components as measured by an observer (i) at rest in the basket and (ii) at rest on the ground.

Dizziness. Our balance is maintained, at least in part, by the endolymph fluid in the inner ear. Spinning displaces this fluid, causing dizziness. Suppose a dancer (or skater) is spinning at a very fast 3.0 revolutions per second about a vertical axis through the center of his head. Although the distance varies from person to person, the inner ear is approximately 7.0 cm from the axis of spin. What is the radial acceleration (in \(\mathrm{m} / \mathrm{s}^{2}\) and in g’s) of the endolymph fluid?

The earth has a radius of \(380 \mathrm{~km}\) and turns around once on its axis in \(24 \mathrm{~h}\). (a) What is the radial acceleration of an object at the earth's equator? Give your answer in \(\mathrm{m} / \mathrm{s}^{2}\) and as a fraction of \(g\). (b) If \(a_{\text {rad }}\) at the equator is greater than \(g\), objects will fly off the earth's surface and into space. (We will see the reason for this in Chapter 5.) What would the period of the earth's rotation have to be for this to occur?

A model of a helicopter rotor has four blades, each 3.40 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at 550 rev/min. (a) What is the linear speed of the blade tip, in m/s? (b) What is the radial acceleration of the blade tip expressed as a multiple of the acceleration of gravity, g?

BIO Pilot Blackout in a Power Dive. A jet plane comes in for a downward dive as shown in Fig. E3.27. The bottom part of the path is a quarter circle with a radius of curvature of \(350 \mathrm{~m}\). According to medical tests, pilots lose consciousness at an acceleration of 5.5????. At what speed (in m/s and in mph) will the pilot black out for this dive? Equation Transcription: Text Transcription: 350 m 5.5g

Problem 28E The radius of the earth’s orbit around the sun (assumed to be circular) is 1.50 X 108 km, and the earth travels around this orbit in 365 days. (a) What is the magnitude of the orbital velocity of the earth, in m/s? (b) What is the radial acceleration of the earth toward the sun, in m/s2? (c) Repeat parts (a) and (b) for the motion of the planet Mercury (orbit radius = 5.79 X 107 km, orbital period = 88.0 days).

A Ferris wheel with radius 14.0 m is turning about a horizontal axis through its center (Fig. E3.29). The linear speed of a passenger on the rim is constant and equal to 7.00 m/s. What are the magnitude and direction of the passenger’s acceleration as she passes through (a) the lowest point in her circular motion? (b) The highest point in her circular motion? (c) How much time does it take the Ferris wheel to make one revolution?

Problem 30E BIO Hypergravity. At its Ames Research Center, NASA uses its large “20-G” centrifuge to test the effects of very large accelerations (“hypergravity”) on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge’s arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5 g. (a) How fast must the astronaut’s head be moving to experience this maximum acceleration? (b) What is the difference between the acceleration of his head and feet if the astronaut is 2.00 m tall? (c) How fast in rpm (rev/min) is the arm turning to produce the maximum sustained acceleration?

A “moving sidewalk” in an airport terminal building moves at 1.0 m/s and is 35.0 m long. If a woman steps on at one end and walks at 1.5 m/s relative to the moving sidewalk, how much time does she require to reach the opposite end if she walks (a) in the same direction the sidewalk is moving? (b) In the opposite direction?

A railroad flatcar is traveling to the right at a speed of \(13.0 \mathrm{~m} / \mathrm{s}\) relative to an observer standing on the ground. Someone is riding a motor scooter on the flatcar (Fig. E3.32). What is the velocity (magnitude and direction) of the motor scooter relative to the flatcar if its velocity relative to the observer on the ground is (a) \(18.0 \mathrm{~m} / \mathrm{s}\) to the left? (c) zero? Equation Transcription: Text Transcription: 13.0 m/s 18.0 m/s

Problem 33E A canoe has a velocity of 0.40 m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50 m/s east relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

Two piers, A and B, are located on a river: B is \(1500 \mathrm{~m}\) downstream from A (Fig. E3.34). Two friends must make round trips from pier A to pier B and return. One rows a boat at a constant speed of 4.00 \(\mathrm{~km} / \mathrm{h}\) relative to the water; the other walks on the shore at a constant speed of \(4.00 \mathrm{~km} / \mathrm{h}\). The velocity of the river is 2.80 km/h in the direction from A to B. How much time does it take each person to make the round trip? Equation Transcription: Text Transcription: 1500 m 4.00km/h 2.80km/h 4.00km/h

Crossing the River I. A river flows due south with a speed of 2.0 m/s. A man steers a motorboat across the river; his velocity relative to the water is 4.2 m/s due east. The river is 800 m wide. (a) What is his velocity (magnitude and direction) relative to the earth? (b) How much time is required to cross the river? (c) How far south of his starting point will he reach the opposite bank?

Problem 36E Crossing the River II. (a) In which direction should the motorboat in head in order to reach a point on the opposite bank directly east from the starting point? (The boat’s speed relative to the water remains 4.2 m/s.) (b) What is the velocity of the boat relative to the earth? (c) How much lime is required to cross the river? Exercise: Crossing the River I. A river flows due south with a speed of 2.0 m/s. A man steers a motorboat across the river; his velocity relative to the water is 4.2 m/s due east. The river is 800 m wide. (a) What is his velocity (magnitude and direction) relative to the earth? (b) How much time is required to cross the river? (c) How far south of his starting point will he reach the opposite bank?

The nose of an ultralight plane is pointed south, and its airspeed indicator shows 35 m/s. The plane is in a 10-m/s wind blowing toward the southwest relative to the earth. (a) In a vector addition diagram, show the relationship of \(\vec{v}_{P / E}\) (the velocity of the plane relative to the earth) to the two given vectors. (b) Letting ???? be east and ???? be north, find the components of \(\vec{v}_{P / E}\). (c) Find the magnitude and direction of \(\vec{v}_{P / E}\). Equation Transcription: Text Transcription: vP/E x y vP/E vP/E

An airplane pilot wishes to fly due west. A wind of 80.0 km/h (about 50 mi/h) is blowing toward the south. (a) If the airspeed of the plane (its speed in still air) is 320.0 km/h (about 200 mi/h), in which direction should the pilot head? (b) What is the speed of the plane over the ground? Illustrate with a vector diagram.

Bird Migration. Canadian geese migrate essentially along a north–south direction for well over a thousand kilometers in some cases, traveling at speeds up to about 100 km/h. If one such bird is flying at 100 km/h relative to the air, but there is a 40 km/h wind blowing from west to east, (a) at what angle relative to the north–south direction should this bird head so that it will be traveling directly southward relative to the ground? (b) How long will it take the bird to cover a ground distance of 500 km from north to south? (Note: Even on cloudy nights, many birds can navigate using the earth’s magnetic field to fix the north–south direction.)

An athlete starts at point A and runs at a constant speed of 6.0 m/s around a circular track 100 m in diameter, as shown in Fig. P3.40. Find the \(x \text { and } y \text {-components }\) of this runner’s average velocity and average acceleration between points (a) A and B, (b) A and C, (c) C and D, and (d) A and A (a full lap). (e) Calculate the magnitude of the runner’s average velocity between A and B. Is his average speed equal to the magnitude of his average velocity? Why or why not? (f) How can his velocity be changing if he is running at constant speed? Equation Transcription: Text Transcription: x and y - components

A faulty model rocket moves in the xy-plane (the positive y-direction is vertically upward). The rocket’s acceleration has components \(a_{x}(t)=\alpha t^{2}\) and \(a_{y}(t)=\beta-\gamma t\), where \(\alpha=2.50\mathrm{\ m}/\mathrm{s}^4,\ \beta=9.00\mathrm{\ m}/\mathrm{s}^2\), and \(\gamma=1.40\mathrm{\ m}/\mathrm{s}^3\). At t = 0 the rocket is at the origin and has velocity \(\vec{\boldsymbol{v}}_{0}=v_{0 x} \hat{\imath}+v_{0 y} \hat{\jmath}\) with \(v_{0x}=1.00\mathrm{\ m}/\mathrm{s}\) and \(v_{0y}=7.00\mathrm{\ m}/\mathrm{s}\). (a) Calculate the velocity and position vectors as functions of time. (b) What is the maximum height reached by the rocket? (c) Sketch the path of the rocket. (d) What is the horizontal displacement of the rocket when it returns to y = 0?

If \(\vec{r}=b t^{2} \hat{\imath}+c t^{3} \hat{j},\) where b and c are positive constants, when does the velocity vector make an angle of \(45.0^{\circ}\) with the \(x\)- and \(y\)-axes? Equation Transcription: Text Transcription: r =bt2î+ct3 ? 45.0°

CALC The position of a dragonfly that is flying parallel to the ground is given as a function of time by \(\vec{r}=\left[2.90 \mathrm{~m}+\left(0.0900 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}\right] \hat{\imath}-\left(0.0150 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3} \hat{\jmath}\) (a) At what value of ???? does the velocity vector of the insect make an angle of 30.0° clockwise from the +????-axis? (b) At the time calculated in part (a), what are the magnitude and direction of the acceleration vector of the insect? Equation Transcription: Text Transcription: r =2.90 m + (0.0900 m/s2)t2î-(0.0150 m/s3)t3?

A small toy airplane is flying in the ????????-plane parallel to the ground. In the time interval \(t=1.00 \mathrm{~s}\), its velocity as a function of time is given by \(\vec{v}=\left(1.20 \mathrm{~m} / \mathrm{s}^{2}\right) t \hat{\imath}+\left[12.0 \mathrm{~m} / \mathrm{s}-\left(2.00 \mathrm{~m} / \mathrm{s}^{2}\right) t\right] \hat{\jmath}\). At what value of ???? is the velocity of the plane perpendicular to its acceleration? Equation Transcription: Text Transcription: t=1.00 s v=(1.20 m/s2)tî+12.0 m/s - (2.00 m/s2)t?

CALC A bird flies in the xy-plane with a velocity vector given by \(\vec{v}=\left(\alpha-\beta t^{2}\right) \hat{\imath}+\gamma t \hat{\jmath} .\). With \(\alpha=2.4 \mathrm{~m} / \mathrm{s}, \beta=1.6 \mathrm{~m} / \mathrm{s}^{3}, \text { and } \gamma=4.0 \mathrm{~m} / \mathrm{s}^{2}\). The positive ????-direction is vertically upward. At ???? = 0 the bird is at the origin. (a) Calculate the position and acceleration vectors of the bird as functions of time. (b) What is the bird’s altitude (????-coordinate) as it flies over ???? = 0 for the first time after ???? = 0? Equation Transcription: Text Transcription: v=(-t2) î+t? alpha=2.4 m/s, =1.6 m/s3, and =4.0 m/s2 t=0 x=0 t=0

CP A test rocket is launched by accelerating it along a \(200.0-\mathrm{m}\) incline at \(1.25 \mathrm{~m} / \mathrm{s}^{2}\) starting from rest at point \(A\) (Fig. P3.47). The incline rises at \(35.0^{\circ}\) above the horizontal, and at the instant the rocket leaves it, its engines turn off and it is subject only to gravity (air resistance can be ignored). Find (a) the maximum height above the ground that the rocket reaches, and (b) the greatest horizontal range of the rocket beyond point \(A\).

Problem 48P Martian Athletics. In the long jump, an athlete launches herself at an angle above the ground and lands at the same height, trying to travel the greatest horizontal distance. Suppose that on earth she is in the air for time T, reaches a maximum height h, and achieves a horizontal distance D. If she jumped in exactly the same way during a competition on Mars. where gMars, is 0.379 or its earth value. find her time in the air, maximum height, and horizontal distance. Express each or these three quantities in terms or its earth value. Air resistance can be neglected on both planets.

Problem 49P Dynamite! A demolition crew uses dynamite to blow an old building apart. Debris from the explosion flies off in all directions and is later found at distances as far as 50 m from the explosion. Estimate the maximum peed al which debris was blown outward by the explosion. Describe any assumptions that you make.

Problem 50P BIO Spiraling Up. Birds of prey typically rise upward on thermals. The paths these birds take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume that a bird completes a circle of radius 6.00 m every 5.00 s and rises vertically at a constant rate of 3.00 m/s. Determine (a) the bird’s speed relative to the ground; (b) the bird’s acceleration (magnitude and direction); and (c) the angle between the bird’s velocity vector and the horizontal.

Problem 51P A sly 1.5-kg monkey and a jungle veterinarian with a blow-gun loaded with a tranquilizer dart are 25 m above the ground in trees 70 m apart. Just as the veterinarian shoots horizontally at the monkey, the monkey drops from the tree in a vain attempt to escape being hit. What must the minimum muzzle velocity of the dart be for the dart to hit the monkey before the monkey reaches the ground?

Problem 52P A movie stuntwoman drops from a helicopter that is 30.0 m above the ground and moving with a constant velocity whose components are 10.0 m/s upward and 15.0 m/s horizontal and toward the south. Ignore air resistance. (a) Where on the ground (relative to the position of the helicopter when she drops) should the stuntwoman have placed foam mats to break her fall? (b) Draw x-t, y-t, vx-t, and vy-t graphs of her motion.

Problem 53P In fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. For practice, a pilot drops a canister of red dye, hoping to hit a target on the ground below. If the plane is flying in a horizontal path 90.0 m above the ground and has a speed of 64.0 m/s (143 mi/h), at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.

A cannon, located 60.0 m from the base of a vertical 25.0-m-tall cliff, shoots a 15-kg shell at \(43.0^{\circ}\) above the horizontal toward the cliff. (a) What must the minimum muzzle velocity be for the shell to clear the top of the cliff? (b) The ground at the top of the cliff is level, with a constant elevation of 25.0 m above the cannon. Under the conditions of part (a), how far does the shell land past the edge of the cliff?

Problem 54P A cannon, located 60.0 m from the base of a vertical 25.0-m-tall cliff, shoots a 15-kg shell at 43.0o above the horizontal toward the cliff. (a) What must the minimum muzzle velocity be for the shell to clear the top of the cliff? (b) The ground at the top of the cliff is level, with a constant elevation of 25.0 m above the cannon. Under the conditions of part (a), how far does the shell land past the edge of the cliff?

As a ship is approaching the dock at \(45.0 \mathrm{~cm} / \mathrm{s}\), an important piece of landing equipment needs to be thrown to it before it can dock. This equipment is thrown at \(\text { q15.0 } \mathrm{m} / \mathrm{s}\) at \(60.0^{\circ}\) above the horizontal from the top of a tower at the edge of the water, 8.75 m above the ship’s deck (Fig. P3.56). For this equipment to land at the front of the ship, at what distance D from the dock should the ship be when the equipment is thrown? Air resistance can be neglected. Equation Transcription: ° Text Transcription: 45.0 cm/s q15.0 m/s 60.0^°

Problem 57P CP CALC A toy rocket is launched with an initial velocity of 12.0 m/s in the horizontal direction from the roof of a 30.0-m-tall building. The rocket’s engine produces a horizontal acceleration of (1.60 m/s3)t, in the same direction as the initial velocity, but in the vertical direction the acceleration is g , down-ward. Ignore air resistance. What horizontal distance does the rocket travel before reaching the ground?

Problem 58P An Errand of Mercy. An airplane is dropping bales of hay to cattle stranded in a blizzard on the Great Plains. The pilot releases the bales at 150 m above the level ground when the plane is flying at 75 m/s in a direction 55° above the horizontal. How far in front of the cattle should the pilot release the hay so that the bales land at the point where the cattle are stranded?

The Longest Home Run. According to the Guinness Book of World Records, the longest home run ever measured was hit by Roy “Dizzy” Carlyle in a minor league game. The ball traveled 188 m (618 ft) before landing on the ground outside the ballpark. (a) Assuming the ball’s initial velocity was in a direction \(45^{\circ}\) above the horizontal and ignoring air resistance, what did the initial speed of the ball need to be to produce such a home run if the ball was hit at a point 0.9 m (3.0 ft) above ground level? Assume that the ground was perfectly flat. (b) How far would the ball be above a fence 3.0 m (10 ft) high if the fence was 116 m (380 ft) from home plate?

A water hose is used to fill a large cylindrical storage tank of diameter D and height 2D. The hose shoots the water at \(45^{\circ}\) above the horizontal from the same level as the base of the tank and is a distance 6D away (Fig. P3.60). For what range of launch speeds \(\left(v_{0}\right)\) will the water enter the tank? Ignore air resistance, and express your answer in terms of D and ????. Equation Transcription: ° Text Transcription: 45° (v_o)

A projectile is being launched from ground level with no air resistance. You want to avoid having it enter a temperature inversion layer in the atmosphere a height h above the ground. (a) What is the maximum launch speed you could give this projectile if you shot it straight up? Express your answer in terms of h and g. (b) Suppose the launcher available shoots projectiles at twice the maximum launch speed you found in part (a). At what maximum angle above the horizontal should you launch the projectile? (c) How far (in terms of h) from the launcher does the projectile in part (b) land?

Kicking a Field Goal. In U.S. football, after a touchdown the team has the opportunity to earn one more point by kicking the ball over the bar between the goal posts. The bar is 10.0 ft above the ground, and the ball is kicked from ground level, 36.0 ft horizontally from the bar (Fig. P3.62). Football regulations are stated in English units, but convert them to SI units for this problem. (a) There is a minimum angle above the ground such that if the ball is launched below this angle, it can never clear the bar, no matter how fast it is kicked. What is this angle? (b) If the ball is kicked at \(45.0^{\circ}\) above the horizontal, what must its initial speed be if it is to just clear the bar? Express your answer in m/s and in km/h.

A grasshopper leaps into the air from the edge of a vertical cliff, as shown in Fig. P3.63. Use information from the figure to find (a) the initial speed of the grasshopper and (b) the height of the cliff. Equation Transcription: ° Text Transcription: 6.74 cm 50.0° 1.06 m

Problem 64P A World Record. In the shot put, a standard track and-field event, a 7.3-kg object (the shot) is thrown by releasing it at approximately 40° over a straight left leg. The world record for distance, set by Randy Barnes in 1990, is 23.11 m. Assuming that Barnes released the shot put at 40.0° from a height of 2.00 m above the ground, with what speed, in m/s and in mph, did he release it?

Look Out! A snowball rolls off a barn roof that slopes downward at an angle of \(40^{\circ}\) (Fig. P3.65). The edge of the roof is \(14.0 \mathrm{~m}\) above the ground, and the snowball has a speed of \(7.00 \mathrm{~m} / \mathrm{s}\) as it rolls off the roof. Ignore air resistance. (a) How far from the edge of the barn does the snowball strike the ground if it doesn't strike anything else while falling? (b) Draw \(x-t, y-t\), \(v_{x}-t\), and \(v_{y}-t\) graphs for the motion in part (a). (c) A man \(1.9 \mathrm{~m}\) tall is standing \(4.0 \mathrm{~m}\) from the edge of the barn. Will he be hit by the snowball?

On the Flying Trapeze. A new circus act is called the Texas Tumblers. Lovely Mary Belle swings from a trapeze, projects herself at an angle of and is supposed to be caught by Joe Bob, whose hands are 6.1 m above and 8.2 m horizontally from her launch point (Fig. P3.66). You can ignore air resistance. (a) What initial speed \(V_{0}\)must Mary Belle have just to reach Joe Bob? (b) For the initial speed calculated in part (a), what are the magnitude and direction of her velocity when Mary Belle reaches Joe Bob? (c) Assuming that Mary Belle has the initial speed calculated in part (a), draw \(x-t, y-t, v_{x_{-}} t \text { and } v_{\gamma}-t\) graphs showing the motion of both tumblers. Your graphs should show the motion up until the point where Mary Belle reaches Joe Bob. (d) The night of their debut performance, Joe Bob misses her completely as she flies past. How far horizontally does Mary Belle travel, from her initial launch point, before landing in the safety net 8.6 m below her starting point? Equation Transcription: 53° Text Transcription: 53° vo x-t, y-t, ux-t and uy

A physics professor did daredevil stunts in his spare time. His last stunt was an attempt to jump across a river on a motorcycle (Fig. P3.67). The takeoff ramp was inclined at \(53.0^{\circ}\), the river was 40.0 m wide, and the far bank was 15.0 m lower than the top of the ramp. The river itself was 100 m below the ramp. You can ignore air resistance. (a) What should his speed have been at the top of the ramp to have just made it to the edge of the far bank? (b) If his speed was only half the value found in part (a), where did he land? Equation Transcription: ° Text Transcription: 53.0^°

Problem 68P A rock is thrown with a velocity v0, at an angle of ?0 from the horizontal, from the roof of a building of height h. Ignore air resistance. Calculate the speed of the rock just before it strikes the ground, and show that this speed is independent of ?0.

Problem 69P A 5500-kg cart carrying a vertical rocket launcher moves to the right at a constant speed of 30.0 m/s along a horizontal track. It launches a 45.0-kg rocket vertically upward with an initial speed of 40.0 m/s relative to the cart. (a) How high will the rocket go? (b) Where, relative to the cart, will the rocket land How far dues the cart move while the rocket is in the air? (d) At what angle relative to the horizontal is the rocket traveling just as it the curl as measured by an observer at rest on the ground? (e) Sketch the rocket’s trajectory been by an observer (i) Stationary on the cart and (ii) ordinary on the ground.

A 2.7-kg ball is thrown upward with an initial speed of 20.0 m/s from the edge of a 45.0-m-high cliff. At the instant the ball is thrown, a woman starts running away from the base of the cliff with a constant speed of 6.00 m/s. The woman runs in a straight line on level ground. Ignore air resistance on the ball. (a) At what angle above the horizontal should the ball be thrown so that the runner will catch it just before it hits the ground, and how far does she run before she catches the ball? (b) Carefully sketch the ball’s trajectory as viewed by (i) a person at rest on the ground and (ii) the runner.

A 76.0-kg boulder is rolling horizontally at the top of a vertical cliff that is 20 m above the surface of a lake, as shown in Fig. P3.71. The top of the vertical face of a dam is located 100 m from the foot of the cliff, with the top of the dam level with the surface of the water in the lake. A level plain is 25 m below the top of the dam. (a) What must be the minimum speed of the rock just as it leaves the cliff so it will travel to the plain without striking the dam? (b) How far from the foot of the dam does the rock hit the plain?

Problem 72P Tossing Your Lunch. Henrietta is jogging on the side-walk at 3.05 m/s on the way to her physics class. Bruce realizes that she forgot her bag of bagels, so he runs to the window, which is 38.0 m above the street level and directly above the sidewalk, to throw the bag to her. He throws it horizontally 9.00 s after she has passed below the window, and she catches it on the run. Ignore air resistance. (a) With what initial speed must Bruce throw the bagels so that Henrietta can catch the bag just before it hits the ground? (b) Where is Henrietta when she catches the bagels?

Two tanks are engaged in a training exercise on level ground. The first tank fires a paint-filled training round with a muzzle speed of 250 m / s at \(10.0^{\circ}\) above the horizontal while advancing toward the second tank with a speed of 15.0 m / s relative to the ground. The second tank is retreating at 35.0 m / s relative to the ground, but is hit by the shell. You can ignore air resistance and assume the shell hits at the same height above ground from which it was fired. Find the distance between the tanks (a) when the round was first fired and (b) at the time of impact.

Problem 74P CP Bang! A student sits atop a platform a distance h above the ground. He throws a large firecracker horizontally with a speed. However, a wind blowing parallel to the ground gives the firecracker a constant horizontal acceleration with magnitude a. As a result, the firecracker reaches the ground directly below the student. Determine the height h in terms of v, a, and g. Ignore the effect of air resistance on the vertical motion.

Problem 75P In a Fourth of July celebration, a firework is launched from ground travel with an initial velocity of 25.0 m/s at 30.0° from the vertical. At its maximum height it explodes in a starburst into many fragments, two of which travel forward initially at 20.0 m/s at ± 53.0 with respect to the horizontal both quantities measured relative to the original firework just before it exploded. With what angles with respect to the horizontal do the two fragments initially move right after the explosion as measured by a spectator standing on the ground?

Problem 76P When it is 145 m above the ground, a rocket traveling vertical by upward at a constant 8.50 m/s relative to the ground launches a secondary rocket at a speed of 12.0 m/s at an angle of 53.0° above the horizontal, both quantities being measured by an astronaut sitting in the rocket. After it is launched the secondary rocket is in free-fall. (a) Just as the secondary rocket is launched, what are the horizontal and vertical components of its velocity relative to (i) the astronaut sitting in the rocket and (ii) Mission Control on the ground? (b) Find the initial speed and launch angle of the secondary rocket as measured by Mission Control. (c) What maximum height above the ground does the secondary rocket reach?

Problem 77P In an action-adventure film, the hero is supposed to throw a grenade from his car. which is going 90.0 km/h, to his enemy’s car, which is going 110 km/h. The enemy’s car is 15.8 m in front of the hero’s when he lets go of the grenade. If the hero throws the grenade so its initial velocity relative to him is at an angle of 45° above the horizontal, what should the magnitude of the initial velocity be? The cars are both traveling in the same direction on a level road. You can ignore air resistance. Find the magnitude of the velocity both relative to the hero and relative to the earth.

A 400.0-m-wide river flows from west to east at \(30.0 \mathrm{~m} / \mathrm{min} \text {. }\). Your boat moves at \(100.0 \mathrm{~m} / \mathrm{min}\) relative to the water no matter which direction you point it. To cross this river, you start from a dock at point A on the south bank. There is a boat landing directly opposite at point B on the north bank, and also one at point C, 75.0 m downstream from B (Fig. P3.78). (a) Where on the north shore will you land if you point your boat perpendicular to the water current, and what distance will you have traveled? (b) If you initially aim your boat directly toward point C and do not change that bearing relative to the shore, where on the north shore will you land? (c) To reach point C: (i) at what bearing must you aim your boat, (ii) how long will it take to cross the river, (iii) what distance do you travel, and (iv) and what is the speed of your boat as measured by an observer standing on the river bank? Equation Transcription: Text Transcription: 30.0 m/min 100.0 m/min

Problem 79P Cycloid. A particle moves in the xy-plane. Its coordinates are given as functions of time by x(t) = R(?t ? sin ?t) y(t) = R(1 ? cos ?t) Where R and ? are constants. (a) Sketch the trajectory of the particle. (This is the trajectory of a point on the rim of a wheel that is rolling at a constant speed on a horizontal surface. The curve traced out by such a point as it moves through space is called a cycloid.) (b) Determine the velocity components and the acceleration components of the particle at any time t. (c) At what times is the particle momentarily at rest? What are the coordinates of the particle at these times? What are the magnitude and direction of the (d) Does the magnitude of the acceleration depend on time? Compare to uniform circular motion.

Problem 80P A projectile is fired from point A at an angle above the horizontal. At its highest point, after having traveled a horizontal distance D from its launch point it suddenly explodes into two identical fragments that travel horizontally with equal but opposite velocities as measured relative to the projectile just before it exploded. If one fragment lands back at point A, how far from A (in terms of D) does the other fragment land?

Problem 81P An airplane pilot sets a compass course due west and maintains an airspeed of 220 km/h. After flying for 0.500 h, she finds herself over a town 120 km west and 20 km south of her starting point. (a) Find the wind velocity (magnitude and direction). (b) If the wind velocity is 40 km/h due south, in what direction should the pilot set her course to travel due west? Use the same airspeed of 220 km/h.

Problem 82P Raindrops. When a train’s velocity is 12.0 m/s east-ward, raindrops that are falling vertically with respect to the earth make traces that are inclined 30.0o to the vertical on the windows of the train. (a) What is the horizontal component of a drop’s velocity with respect to the earth? With respect to the train? (b) What is the magnitude of the velocity of the raindrop with respect to the earth? With respect to the train?

Problem 83P In a World Cup soccer match, Juan is running due north toward the goal with a speed of 8.00 m/s relative to the ground. A teammate passes the ball to him. The ball has a speed of 12.0 m/s and is moving in a direction 37.0o east of north, relative to the ground. What are the magnitude and direction of the ball’s velocity relative to Juan?

Problem 84P An elevator is moving upward at a constant speed of 2.50 m/s. A bolt in the elevator ceiling 3.00 m above the elevator floor works loose and falls. (a) How long does it take for the bolt to fall to the elevator floor? What is the speed of the bolt just as it hits the elevator floor (b) according to an observer in the elevator? (c) According to an observer standing on one of the floor landings of the building? (d) According to the observer in part (c), what distance did the bolt travel between the ceiling and the floor of the elevator?

Suppose the elevator in Problem 3.84 starts from rest and maintains a constant upward acceleration of \(4.00\mathrm{\ m}/\mathrm{s}^2\), and the bolt falls out the instant the elevator begins to move. (a) How long does it take for the bolt to reach the floor of the elevator? (b) Just as it reaches the floor, how fast is the bolt moving according to an observer (i) in the elevator? (ii) Standing on the floor landings of the building? (c) According to each observer in part (b), how far has the bolt traveled between the ceiling and floor of the elevator?

Problem 86P Two soccer players, Mia and Alice, are running as Alice passes the ball to Mia. Mia is running due north with a speed of 6.00 m/s. The velocity of the ball relative to Mia is 5.00 m/s in a direction 30.0 east of south. What are the magnitude and direction of the velocity of the ball relative to the ground?

Problem 87P Projectile Motion on an Incline. Refer to the Bridging Problem in chapter 3. (a) An archer on ground that has a constant upward slope of 30.0° aims at a target 60.0 in farther up the incline. The arrow in the bow and the bull’s-eye at the center of the target are each 1.50 m above the ground. The initial velocity of the arrow just after it leaves the bow has magnitude 32.0 m/s. At what angle above the horizontal should the archer aim to hit the bull ’s–eye? If there are two such angles, calculate the smaller of the two. You might have to solve the equation for the angle by itcration — that is by trial and error. How does the angle compare to that required when the ground in level with slope? (b) Repeat the problem in ground that has a constant downward slope of 30.0°.

Problem 88CP CALC A projectile thrown from a point P moves in such a way that its distance from P is always increasing. Find the maximum angle above the horizontal with which the projectile could have been thrown. Ignore air resistance.

Problem 89CP Two students are canoeing on a river. While heading upstream, they accidentally drop an empty bottle overboard. They then continue paddling for 60 minutes, reaching a point 2.0 km farther upstream. At this point they realize that the bottle is missing and, driven by ecological awareness, they turn around and head downstream. They catch up with and retrieve the bottle (which has been moving along with the current) 5.0 km downstream from the turnaround point. (a) Assuming a constant paddling effort throughout, how fast is the river flowing? (b) What would the canoe speed in a still lake be for the same paddling effort?

Problem 90CP CP A rocket designed to place small payloads into orbit is carried to an altitude of 12.0 km above sea level by a converted airliner. When the airliner is flying in a straight line at a constant speed of 850 km/h, the rocket is dropped. After the drop, the air-liner maintains the same altitude and speed and continues to fly in a straight line. The rocket falls for a brief time, after which its rocket motor turns on. Once that motor is on, the combined effects of thrust and gravity give the rocket a constant acceleration of magnitude 3.00 g directed at an angle of 30.0o above the horizontal. For safety, the rocket should be at least 1.00 km in front of the airliner when it climbs through the airliner’s altitude. Your job is to determine the minimum time that the rocket must fall before its engine starts. Ignore air resistance. Your answer should include (i) a diagram showing the flight paths of both the rocket and the airliner, labeled at several points with vectors for their velocities and accelerations; (ii) an x-t graph showing the motions of both the rocket and the airliner; and (iii) a y-t graph showing the motions of both the rocket and the airliner. In the diagram and the graphs, indicate when the rocket is dropped, when the rocket motor turns on, and when the rocket climbs through the altitude of the airliner.

Problem 12DQ In uniform circular motion, the acceleration is perpendicular to the velocity at every instant. Is this true when the motion is not uniform—that is, when the speed is not constant?