Answer: In uniform circular motion, the acceleration is

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.

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?

A projectile moves in a parabolic path without air resistance. Is there any point at which a S is parallel to v S ? Perpendicular to v S ? Explain

A book slides off a horizontal tabletop. As it leaves the tables edge, the book has a horizontal velocity of magnitude v0. The book strikes the floor in time t. If the initial velocity of the book is doubled to 2v0, what happens to (a) the time the book is in the air, (b) the horizontal distance the book travels while it is in the air, and (c) the speed of the book just before it reaches the floor? In particular, does each of these quantities stay the same, double, or change in another way? Explain

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

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?

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 6 a0 6 90

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 2 0 >g?

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

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

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?

In uniform circular motion, the acceleration is perpendicular to the velocity at every instant. Is this true when the motion is not uniformthat is, when the speed is not constant?

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?

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

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

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 stones speed v as a function of time t while it is in the air?

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 \(t_{2}=3.0 \mathrm{\ s}\). 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 t1 = 0. For the time interval from t1 = 0 to t2 = 12.0 s, the rhinos 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 web page designer creates an animation in which a dot on a computer screen has position \(\overrightarrow{\boldsymbol{r}}=\left[4.0 \mathrm{\ cm}+\left(2.5 \mathrm{\ cm} / \mathrm{s}^{2}\right) t^{2}\right] \hat{\imath}+(5.0 \mathrm{\ cm} / \mathrm{s}) t \hat{\jmath}\) (a) Find the magnitude and direction of the dot’s average velocity between t = 0 and t = 2.0 s.(b) Find the magnitude and direction of the instantaneous velocity at t = 0, t = 1.0 s, and t = 2.0. (c) Sketch the dot’s trajectory from t = 0 to t = 2.0 s, and show the velocities calculated in part (b).

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 is the magnitude and direction of the squirrel’s velocity?

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 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>s 2 and direction 31.0 measured from the +x@axis toward the +y@axis. At t2 = 20.0 s, (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 t1 and t2. How do these two vectors differ?

The coordinates of a bird flying in the xy-plane are given by x1t2 = at and y1t2 = 3.0 m - bt 2 , where a = 2.4 m>s and b = 1.2 m>s 2 . (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 birds 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 birds speed increasing, decreasing, or not changing? Is the bird turning? If so, in what direction?

The coordinates of a bird flying in the xy-plane are given by x1t2 = at and y1t2 = 3.0 m - bt 2 , where a = 2.4 m>s and b = 1.2 m>s 2 . (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 birds 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 birds speed increasing, decreasing, or not changing? Is the bird turning? If so, in what direction?

A physics book slides off a horizontal tabletop with a speed of 1.10 m>s. It strikes the floor in 0.480 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 books 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

. 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?

Crickets Chirpy and Milada jump from the top of a vertical cliff. Chirpy drops downward and reaches the ground in 2.70 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? Ignore air resistance

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

Leaping the River I. During a storm, a car traveling on a level horizontal road comes upon a bridge that has washed out. The driver must get to the other side, so he decides to try leaping the river with his car. The side of the road the car is on is 21.3 m above the river, while the opposite side is only 1.8 m above the river. The river itself is a raging torrent 48.0 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?

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 40.0 m>s 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 above the horizontal. 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 baseballs velocity at each of the two times calculated in part (a). (c) What are the magnitude and direction of the baseballs 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 m/s, 51.0° above the horizontal. The shot hits the ground 2.08 s later. 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 \(x-t, y-t, v_{x}-t, \text { and } v_{y}-t\) graphs for the motion.

Win the Prize. In a carnival booth, you can 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 above the horizontal, the coin will land in the dish. 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?

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 a 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 a. (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 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 33.0 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.

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. That person sees the stone hit the ground 5.00 s after it was thrown. Assume that the balloon continues its downward descent with the same constant speed of 20.0 m>s. (a) How high is the balloon when the rock is thrown? (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.

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 earth’s equator? Give your answer in \(\mathrm{m} / \mathrm{s}^{2}\) and as a fraction of g. (b) If \(a_{\mathrm{rad}}\) 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?

Dizziness. Our balance is maintained, at least in part, by the endolymph fluid in the inner ear. Spinning displaces this fluid, causing dizziness. Suppose that a skater is spinning very fast at 3.0 revolutions per second about a vertical axis through the center of his head. Take the inner ear to be approximately 7.0 cm from the axis of spin. (The distance varies from person to person.) What is the radial acceleration (in m>s 2 and in gs) of the endolymph fluid?

Pilot Blackout in a Power Dive. A jet plane comes in for a downward dive as shown in Fig. E3.25. The bottom part of the path is a quarter circle with a radius of curvature of 280 m. According to medical tests, pilots will lose consciousness when they pull out of a dive at an upward acceleration greater than 5.5g. At what speed (in m>s and in mph) will the pilot black out during this dive?

. 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 g?

A Ferris wheel with radius 14.0 m is turning about a horizontal axis through its center (Fig. E3.27). The linear speed of a passenger on the rim is constant and equal to 6.00 m>s. What are the magnitude and direction of the passengers acceleration as she passes through (a) the lowest point in her circular motion and (b) the highest point in her circular motion? (c) How much time does it take the Ferris wheel to make one revolution?

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 \(\mathrm {m/s}^2\)? (c) Repeat parts (a) and (b) for the motion of the planet Mercury (orbit radius \(= 5.79 \times 10^7 \ \mathrm {km}\), orbital period = 88.0 days).

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 12.5g. (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 railroad flatcar is traveling to the right at a speed of 13.0 m>s relative to an observer standing on the ground. Someone is riding a motor scooter on the flatcar (Fig. E3.30). What is the velocity (magnitude and direction) of the scooter relative to the flatcar if the scooters velocity relative to the observer on the ground is (a) 18.0 m>s to the right? (b) 3.0 m>s to the left? (c) zero?

A moving sidewalk in an airport terminal 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 it take her to reach the opposite end if she walks (a) in the same direction the sidewalk is moving? (b) In the opposite direction?

Two piers, A and B, are located on a river; B is 1500 m downstream from A (Fig. E3.32). 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?

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

The nose of an ultralight plane is pointed due 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 \(\overrightarrow{\boldsymbol{v}}_{\mathrm{P} / \mathrm{E}}\) (the velocity of the plane relative to the earth) to the two given vectors. (b) Let x be east and y be north, and find the components of \(\overrightarrow{\boldsymbol{v}}_{\mathrm{P} / \mathrm{E}}\). (c) Find the magnitude and direction of \(\overrightarrow{\boldsymbol{v}}_{\mathrm{P} / \mathrm{E}}\).

Crossing the River I. A river flows due south with a speed of 2.0 m>s. You steer a motorboat across the river; your velocity relative to the water is 4.2 m>s due east. The river is 500 m wide. (a) What is your velocity (magnitude and direction) relative to the earth? (b) How much time is required to cross the river? (c) How far south of your starting point will you reach the opposite bank?

Crossing the River II. (a) In which direction should the motorboat in Exercise 3.35 head to reach a point on the opposite bank directly east from your starting point? (The boats 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 time is required to cross the river?

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

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? Draw a vector diagram.

A rocket is fired at an angle from the top of a tower of height h0 = 50.0 m. Because of the design of the engines, its position coordinates are of the form x1t2 = A + Bt2 and y1t2 = C + Dt3 , where A, B, C, and D are constants. The acceleration of the rocket 1.00 s after firing is a S = 14.00nd + 3.00ne2 m>s 2 . 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?

A faulty model rocket moves in the xy-plane (the positive y-direction is vertically upward). The rockets acceleration has components ax1t2 = at 2 and ay1t2 = b - gt, where a = 2.50 m>s 4 , b = 9.00 m>s 2 , and g = 1.40 m>s 3 . At t = 0 the rocket is at the origin and has velocity v S 0 = v0xnd + v0yne with v0x = 1.00 m>s and v0y = 7.00 m>s. (a) Calculate the velocity and position vectors as functions of time. (b) What is the maximum height reached by the rocket? (c) What is the horizontal displacement of the rocket when it returns to y = 0?

If \(\vec{r}=b t^2 \hat{\imath}+c t^3 \hat{\jmath}\), 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?

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 t does the velocity vector of the dragonfly make an angle of \(30.0^{\circ}\) clockwise from the +x-axis? (b) At the time calculated in part (a), what are the magnitude and direction of the dragonfly's acceleration vector?

A test rocket starting from rest at point A is launched by accelerating it along a 200.0-m incline at 1.90 m>s 2 (Fig. P3.43). The incline rises at 35.0 above the horizontal, and at the instant the rocket leaves it, the engines turn off and the rocket is subject to gravity only (ignore air resistance). Find (a) the maximum height above the ground that the rocket reaches, and (b) the rockets greatest horizontal range beyond point A

A bird flies in the xy-plane with a velocity vector given by v S = 1a - bt 2 2nd + gt ne, with a = 2.4 m>s, b = 1.6 m>s 3 , and g = 4.0 m>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 birds altitude (y-coordinate) as it flies over x = 0 for the first time after t = 0?

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?

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 birds speed relative to the ground; (b) the birds acceleration (magnitude and direction); and (c) the angle between the birds velocity vector and the horizontal

. 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 1143 mi>h2, at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.

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 \(x-t, y-t, v_x-t\), and \(v_y-t\) graphs of her motion.

An airplane is flying with a velocity of 90.0 m>s at an angle of 23.0 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? 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 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?

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 11.60 m>s 3 2t, in the same direction as the initial velocity, but in the vertical direction the acceleration is g, downward. Ignore air resistance. What horizontal distance does the rocket travel before reaching the ground?

An important piece of landing equipment must be thrown to a ship, which is moving at 45.0 cm>s, before the ship can dock. This equipment is thrown at 15.0 m>s at 60.0 above the horizontal from the top of a tower at the edge of the water, 8.75 m above the ships deck (Fig. P3.52). 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? Ignore air resistance.

The Longest Home Run. According to Guinness World Records, the longest home run ever measured was hit by Roy DizzyCarlyle in a minor league game. The ball traveled 188 m (618 ft) before landing on the ground outside the ballpark. (a) If the balls initial velocity was in a direction 45 above the horizontal, 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? Ignore air resistance, and 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?

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?

A baseball thrown at an angle of 60.0 above the horizontal strikes a building 18.0 m away at a point 8.00 m above the point from which it is thrown. Ignore air resistance. (a) Find the magnitude of the balls initial velocity (the velocity with which the ball is thrown). (b) Find the magnitude and direction of the velocity of the ball just before it strikes the building.

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. P3.56). For what range of launch speeds 1v02 will the water enter the tank? Ignore air resistance, and express your answer in terms of D and g.

A grasshopper leaps into the air from the edge of a vertical cliff, as shown in Fig. P3.57. Find (a) the initial speed of the grasshopper and (b) the height of the cliff.

Kicking an Extra Point. In Canadian 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. P3.58). 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 just to clear the bar? Express your answer in m>s and in km>h.

Look Out! A snowball rolls off a barn roof that slopes downward at an angle of 40 (Fig. P3.59). The edge of the roof is 14.0 m above the ground, and the snowball has a speed of 7.00 m>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 doesnt strike anything else while falling? (b) Draw x-t, y-t, vx@t, and vy@t 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 the snowball hit him?

A boy 12.0 m above the ground in a tree throws a ball for his dog, who is standing right below the tree and starts running the instant the ball is thrown. If the boy throws the ball horizontally at 8.50 m>s, (a) how fast must the dog run to catch the ball just as it reaches the ground, and (b) how far from the tree will the dog catch the ball?

Suppose that the boy in Problem 3.60 throws the ball upward at 60.0 above the horizontal, but all else is the same. Repeat parts (a) and (b) of that problem

A rock is thrown with a velocity v0, at an angle of a0 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 a0

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. P3.63). The takeoff ramp was inclined at 53.0, 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. 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 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 rock is rolling horizontally at the top of a vertical cliff that is 20 m above the surface of a lake (Fig. P3.65). 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 that it will reach 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 jogging on the sidewalk 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?

A cart carrying a vertical missile launcher moves horizontally at a constant velocity of 30.0 m>s to the right. It launches a rocket vertically upward. The missile has an initial vertical velocity of 40.0 m>s relative to the cart. (a) How high does the rocket go? (b) How far does the cart travel while the rocket is in the air? (c) Where does the rocket land relative to the cart?

A firefighting crew uses a water cannon that shoots water at 25.0 m>s at a fixed angle of 53.0 above the horizontal. The firefighters want to direct the water at a blaze that is 10.0 m above ground level. How far from the building should they position their cannon? There are two possibilities; can you get them both? (Hint: Start with a sketch showing the trajectory of the water.)

In the middle of the night you are standing a horizontal distance of 14.0 m from the high fence that surrounds the estate of your rich uncle. The top of the fence is 5.00 m above the ground. You have taped an important message to a rock that you want to throw over the fence. The ground is level, and the width of the fence is small enough to be ignored. You throw the rock from a height of 1.60 m above the ground and at an angle of \(56.0^\mathrm o\) above the horizontal. (a) What minimum initial speed must the rock have as it leaves your hand to clear the top of the fence? (b) For the initial velocity calculated in part (a), what horizontal distance beyond the fence will the rock land on the ground?

The coordinates of a bird flying in the \(x y\)-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 \(\beta=1.2 \mathrm{~m} / \mathrm{s}^2\). (a) Sketch the path of the bird between \(t=0\) and \(t=2.0 \mathrm{~s}\). (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 \(t=2.0 \mathrm{~s}\). (d) Sketch the velocity and acceleration vectors at \(t=2.0 \mathrm{~s}\). At this instant, is the bird's speed increasing, decreasing, or not changing? Is the bird turning? If so, in what direction?

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

Raindrops. When a trains velocity is 12.0 m>s eastward, raindrops that are falling vertically with respect to the earth make traces that are inclined 30.0 to the vertical on the windows of the train. (a) What is the horizontal component of a drops 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 east of north, relative to the ground. What are the magnitude and direction of the balls velocity relative to Juan?

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?

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 30.0o east of south. What are the magnitude and direction of the velocity of the ball relative to the ground?

A spring-gun projects a small rock from the ground with speed v0 at an angle u0 above the ground. You have been asked to determine v0. From the way the spring-gun is constructed, you know that to a good approximation v0 is independent of the launch angle. You go to a level, open field, select a launch angle, and measure the horizontal distance the rock travels. You use g = 9.80 m>s 2 and ignore the small height of the end of the spring-guns barrel above the ground. Since your measurement includes some uncertainty in values measured for the launch angle and for the horizontal range, you repeat the measurement for several launch angles and obtain the results given in Fig. 3.76. You ignore air resistance because there is no wind and the rock is small and heavy. (a) Select a way to represent the data well as a straight line. (b) Use the slope of the best straight-line fit to your data from part (a) to calculate v0. (c) When the launch angle is 36.9o, what maximum height above the ground does the rock reach?

You have constructed a hair-spray-powered potato gun and want to find the muzzle speed \(v_{0}\) of the potatoes, the speed they have as they leave the end of the gun barrel. You use the same amount of hair spray each time you fire the gun, and you have confirmed by repeated firings at the same height that the muzzle speed is approximately the same for each firing. You climb on a microwave relay tower (with permission, of course) to launch the potatoes horizontally from different heights above the ground. Your friend measures the height of the gun barrel above the ground and the range R of each potato. You obtain the following data: Each of the values of h and R has some measurement error: The muzzle speed is not precisely the same each time, and the barrel isn’t precisely horizontal. So you use all of the measurements to get the best estimate of \(v_{0}\). No wind is blowing, so you decide to ignore air resistance. You use \(g=9.80\mathrm{\ m}/\mathrm{s}^2\) in your analysis. (a) Select a way to represent the data well as a straight line. (b) Use the slope of the best-fit line from part (a) to calculate the average value of \(v_{0}\). (c) What would be the horizontal range of a potato that is fired from ground level at an angle of \(30.0^{\circ}\) above the horizontal? Use the value of \(v_{0}\) that you calculated in part (b).

You are a member of a geological team in Central Africa. Your team comes upon a wide river that is flowing east. You must determine the width of the river and the current speed (the speed of the water relative to the earth). You have a small boat with an outboard motor. By measuring the time it takes to cross a pond where the water isnt flowing, you have calibrated the throttle settings to the speed of the boat in still water. You set the throttle so that the speed of the boat relative to the river is a constant 6.00 m>s. Traveling due north across the river, you reach the opposite bank in 20.1 s. For the return trip, you change the throttle setting so that the speed of the boat relative to the water is 9.00 m>s. You travel due south from one bank to the other and cross the river in 11.2 s. (a) How wide is the river, and what is the current speed? (b) With the throttle set so that the speed of the boat relative to the water is 6.00 m>s, what is the shortest time in which you could cross the river, and where on the far bank would you land?

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.

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?

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 that 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 above the horizontal. For safety, the rocket should be at least 1.00 km in front of the airliner when it climbs through the airliners 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.

The experiment is designed so that the seeds move no more than 0.20 mm between photographic frames. What minimum frame rate for the high-speed camera is needed to achieve this? (a) 250 frames>s; (b) 2500 frames>s; (c) 25,000 frames>s; (d) 250,000 frames>s

About how long does it take a seed launched at 90 at the highest possible initial speed to reach its maximum height? Ignore air resistance. (a) 0.23 s; (b) 0.47 s; (c) 1.0 s; (d) 2.3 s

If a seed is launched at an angle of 0 with the maximum initial speed, how far from the plant will it land? Ignore air resistance, and assume that the ground is flat. (a) 20 cm; (b) 93 cm; (c) 2.2 m; (d) 4.6 m.

A large number of seeds are observed, and their initial launch angles are recorded. The range of projection angles is found to be -51 to 75, with a mean of 31. Approximately 65% of the seeds are launched between 6 and 56. (See W. J. Garrison et al., Ballistic seed projection in two herbaceous species, Amer. J. Bot., Sept. 2000, 87:9, 125764.) Which of these hypotheses is best supported by the data? Seeds are preferentially launched (a) at angles that maximize the height they travel above the plant; (b) at angles below the horizontal in order to drive the seeds into the ground with more force; (c) at angles that maximize the horizontal distance the seeds travel from the plant; (d) at angles that minimize the time the seeds spend exposed to the air.