The range of a cannonball fired horizontally from a cliff

Can the magnitude of the displacement of a particle be less than the distance traveled by the particle along its path? Can its magnitude be more than the distance traveled? Explain.

Give an example in which the distance traveled is a significant amount, yet the corresponding displacement is zero. Can the reverse be true? If so, give an example.

What is the average velocity of a batter who hits a home run (from when he hits the ball to when he touches home plate after rounding the bases)?

A baseball is hit so its initial velocity upon leaving the bat makes an angle of \(30^{\circ}\) above the horizontal. It leaves that bat at a height of \(1.0 \mathrm{~m}\) above the ground and lands untouched for a single. During its flight, from just after it leaves the bat to just before it hits the ground, describe how the angle between its velocity and acceleration vectors changes. Neglect any effects due to air resistance.

If an object is moving toward the west at some instant, in what direction is its acceleration? (a) north, (b) east, (c) west, (d) south, (e) may be any direction.

Two astronauts are working on the lunar surface to install a new telescope. The acceleration due to gravity on the moon is only 1.64 m/s2. One astronaut tosses a wrench to the other astronaut, but the speed of throw is excessive and the wrench goes over her colleagues head. When the wrench is at the highest point of its trajectory, (a) its velocity and acceleration are both zero, (b) its velocity is zero but its acceleration is nonzero, (c) its velocity is nonzero but its acceleration is zero, (d) its velocity and acceleration are both nonzero, (e) insufficient information is given to choose between any of the previous choices.

The velocity of a particle is directed toward the east while the acceleration is directed toward the northwest, as shown in Figure 3-27. The particle is (a) speeding up and turning toward the north, (b) speeding up and turning toward the south, (c) slowing down and turning toward the north, (d) slowing down and turning toward the south, (e) maintaining constant speed and turning toward the south.

Assume you know the position vectors of a particle at two points on its path, one earlier and one later. You also know the time it took the particle to move from one point to the other. Then, you can compute the particles (a) average velocity, (b) average acceleration, (c) instantaneous velocity, and (d) instantaneous acceleration.

Consider the path of a moving particle. (a) How is the velocity vector related geometrically to the path of the particle? (b) Sketch a curved path and draw the velocity vector for the particle for several positions along the path.

The acceleration of a car is zero when it is (a) turning right at a constant speed, (b) driving up a long straight incline at constant speed, (c) traveling over the crest of a hill at constant speed, (d) bottoming out at the lowest point of a valley at constant speed, (e) speeding up as it descends a long straight decline.

Give examples of motion in which the directions of the velocity and acceleration vectors are (a) opposite, (b) the same, and (c) mutually perpendicular.

How is it possible for a particle moving at constant speed to be accelerating? Can a particle with constant velocity be accelerating at the same time?

Imagine throwing a dart straight upward so that it sticks into the ceiling. After it leaves your hand, it steadily slows down as it rises before it sticks. (a) Draw the darts velocity vector at times t1 and t2, where t1 and t2 occur after it leaves your hand but before it hits the ceiling and t2 _ t1 is small. From your drawing, find the direction of the change in velocity , and thus the direction of the acceleration vector. (b) After it has stuck in the ceiling for a few seconds, the dart falls down to the floor. As it falls it speeds up, of course, until it hits the floor. Repeat Part (a) to find the direction of its acceleration vector as it falls. (c) Now imagine tossing the dart horizontally. What is the direction of its acceleration vector after it leaves your hand, but before it strikes the floor?

As a bungee jumper approaches the lowest point in her descent, the rubber cord holding her stretches and she loses speed as she continues to move downward. Assuming that she is dropping straight down, make a motion diagram to find the direction of her acceleration vector as she slows down by drawing her velocity vectors at times t1 and t2, where t1 and t2 are two instants during the portion of her descent in which she is losing speed and t2 _ t1 is small. From your drawing find the direction of the change in velocity , and thus the direction of the acceleration vector.

After reaching the lowest point in her jump at time tlow, a bungee jumper moves upward, gaining speed for a short time until gravity again dominates her motion. Draw her velocity vectors at times t1 and t2, where t2 _ t1 is small and t1 tlow t2. From your drawing find the direction of the change in velocity , and thus the direction of the acceleration vector at time tlow.

A river is 0.76 km wide. The banks are straight and parallel (Figure 3-28). The current is 4.0 km/h and is parallel to the banks. A boat has a maximum speed of 4.0 km/h in still water. The pilot of the boat wishes to go on a straight line from Ato B, where the line AB is perpendicular to the banks. The pilot should (a) head directly across the river, (b) head 53 upstream from the line AB, (c) head 37 upstream from the line AB, (d) give upthe trip from A to B is not possible with a boat of this limited speed, (e) do none of the above.

During a heavy rain, the drops are falling at a constant velocity and at an angle of 10 west of the vertical. You are walking in the rain and notice that only the top surfaces of your clothes are getting wet. In what direction are you walking? Explain.

In Problem 17, what is your walking speed if the speed of the drops relative to the ground is 5.2 m/s?

True or false (ignore any effects due to air resistance): (a) When a projectile is fired horizontally, it takes the same amount of time to reach the ground as an identical projectile dropped from rest from the same height. (b) When a projectile is fired from a certain height at an upward angle, it takes longer to reach the ground than does an identical projectile dropped from rest from the same height. (c) When a projectile is fired horizontally from a certain height, it has a higher speed upon reaching the ground than does an identical projectile dropped from rest from the same height.

A projectile is fired at 35 above the horizontal. Any effects due to air resistance are negligible. At the highest point in its trajectory, its speed is 20 m/s. The initial velocity had a horizontal component of (a) 0, (b) (20 m/s) cos 35, (c) (20 m/s) sin 35, (d) (20 m/s)/cos 35, (e) 20 m/s.

A projectile is fired at 35 above the horizontal. Any effects due to air resistance are negligible. The initial velocity of the projectile in Problem 20 has a vertical component that is (a) less than 20 m/s, (b) greater than 20 m/s, (c) equal to 20 m/s, (d) cannot be determined from the data given.

A projectile is fired at 35 above the horizontal. Any effects due to air resistance are negligible. The projectile lands at the same elevation of launch, so the vertical component of the impact velocity of the projectile is (a) the same as the vertical component of its initial velocity in both magnitude and sign, (b) the same as the vertical component of its initial velocity in magnitude but opposite in sign, (c) less than the vertical component of its initial velocity in magnitude but with the same sign, (d) less than the vertical component of its initial velocity in magnitude but with the opposite sign.

Figure 3-29 represents the parabolic trajectory of a projectile going from Ato E. Air resistance is negligible. What is the direction of the acceleration at point B? (a) up and to the right, (b) down and to the left, (c) straight up, (d) straight down, (e) the acceleration of the ball is zero.

Figure 3-29 represents the trajectory of a projectile going from A to E. Air resistance is negligible. (a) At which point(s) is the speed the greatest? (b) At which point(s) is the speed the least? (c) At which two points is the speed the same? Is the velocity also the same at these points?

True or false: (a) If an objects speed is constant, then its acceleration must be zero. (b) If an objects acceleration is zero, then its speed must be constant. (c) If an objects acceleration is zero, its velocity must be constant. (d) If an objects speed is constant, then its velocity must be constant. (e) If an objects velocity is constant, then its speed must be constant.

The initial and final velocities of a particle are as shown in Figure 3-30. Find the direction of the average acceleration.

The automobile path shown in Figure 3-31 is made up of straight lines and arcs of circles. The automobile starts from rest at point A. After it reaches point B, it travels at constant speed until it reaches point E. It comes to rest at point F. (a) At the middle of each segment (AB, BC, CD, DE, and EF), what is the direction of the velocity vector? (b) At which of these points does the automobile have a nonzero acceleration? In those cases, what is the direction of the acceleration? (c) How do the magnitudes of the acceleration compare for segments BC and DE?

Two cannons are pointed directly toward each other, as shown in Figure 3-32. When fired, the cannonballs will follow the trajectories shownP is the point where the trajectories cross each other. If we want the cannonballs to hit each other, should the gun crews fire cannon A first, cannon B first, or should they fire simultaneously? Ignore any effects due to air resistance.

Galileo wrote the following in his Dialogue concerning the two world systems: Shut yourself up . . . in the main cabin below decks on some large ship, and . . . hang up a bottle that empties drop by drop into a wide vessel beneath it. When you have observed [this] carefully . . . have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that . . . . The droplets will fall as before into the vessel beneath without dropping towards the stern, although while the drops are in the air the ship runs many spans. Explain this quotation.

A man swings a stone attached to a rope in a horizontal circle at constant speed. Figure 3-33 represents the path of the rock looking down from above. (a) Which of the vectors could represent the velocity of the stone? (b) Which could represent the acceleration?

True or false: (a) An object cannot move in a circle unless it has centripetal acceleration. (b) An object cannot move in a circle unless it has tangential acceleration. (c) An object moving in a circle cannot have a variable speed. (d) An object moving in a circle cannot have a constant velocity.

Using a motion diagram, find the direction of the acceleration of the bob of a pendulum when the bob is at a point where it is just reversing its direction.

CONTEXT-RICH During your rookie bungee jump, your friend records your fall using a camcorder. By analyzing it frame by frame, he finds that the y component of your velocity is (recorded every 1/20 of a second) as follows: t (s) 12.05 12.10 12.15 12.20 12.25 12.30 12.35 12.40 12.45 vy _0.78 _0.69 _0.55 _0.35 _0.10 0.15 0.35 0.49 0.53 ( ) (a) Draw a motion diagram. Use it to find the direction and relative magnitude of your average acceleration for each of the eight successive 0.050-s time intervals in the table. (b) Comment on how the y component of your acceleration does or does not vary in sign and magnitude as you reverse your direction of motion. SSM m/s

CONTEXT-RICH Estimate the speed in mph with which water comes out of a garden hose using your past observations of water coming out of garden hoses and your knowledge of projectile motion.

CONTEXT-RICH You won a contest to spend a day with a baseball team during spring training. You are allowed to try to hit some balls thrown by a pitcher. Estimate the acceleration during the hit of a fastball thrown by a major league pitcher when you hit the ball squarelystraight back at the pitcher. You will need to make reasonable choices for ball speeds, both just before and just after the ball is hit, and of the contact time of the ball with the bat.

Estimate how far you can throw a ball if you throw it (a) horizontally while standing on level ground, (b) at _ _ 45 above horizontal while standing on level ground, (c) horizontally from the top of a building 12 m high, (d) at _ _ 45 above horizontal from the top of a building 12 m high. Ignore any effects due to air resistance.

In 1978, Geoff Capes of Great Britain threw a heavy brick a horizontal distance of 44.5 m. Find the approximate speed of the brick at the highest point of its flight, neglecting any effects due to air resistance. Assume the brick landed at the same height it was launched.

A wall clock has a minute hand with a length of 0.50 m and an hour hand with a length of 0.25 m. Take the center of the clock as the origin, and use a Cartesian coordinate system with the positive x axis pointing to 3 oclock and the positive y axis pointing to 12 oclock. Using unit vectors and , express the position vectors of the tip of the hour hand ( ) and the tip of the minute hand ( ) when the clock reads (a) 12:00, (b) 3:00, (c) 6:00, (d) 9:00.

In Problem 38, find the displacements of the tip of each hand (that is, and ) when the time advances from 3:00 P.M. to 6:00 P.M.

In Problem 38, write the vector that describes the displacement of a fly if it quickly goes from the tip of the minute hand to the tip of the hour hand at 3:00 P.M.

CONCEPTUAL, APPROXIMATION A bear, awakening from winter hibernation, staggers directly northeast for 12 m and then due east for 12 m. Show each displacement graphically and graphically determine the single displacement that will take the bear back to her cave to continue her hibernation.

Ascout walks 2.4 km due east from camp, then turns left and walks 2.4 km along the arc of a circle centered at the campsite, and finally walks 1.5 km directly toward the camp. (a) How far is the scout from camp at the end of his walk? (b) In what direction is the scouts position relative to the campsite? (c) What is the ratio of the final magnitude of the displacement to the total distance walked.

The faces of a cubical storage cabinet in your garage have 3.0-m-long edges that are parallel to the xyz coordinate planes. The cube has one corner at the origin. A cockroach, on the hunt for crumbs of food, begins at that corner and walks along three edges until it is at the far corner. (a) Write the roachs displacement using the set of , , and unit vectors, and (b) find the magnitude of its displacement. SSM

CONTEXT-RICH You are the navigator of a ship at sea. You receive radio signals from two transmitters A and B, which are 100 km apart, one due south of the other. The direction finder shows you that transmitter A is at a heading of 30 south of east from the ship, while transmitter B is due east. Calculate the distance between your ship and transmitter B.

A stationary radar operator determines that a ship is 10 km due south of him. An hour later the same ship is 20 km due southeast. If the ship moved at constant speed and always in the same direction, what was its velocity during this time?

A particles position coordinates (x, y) are (2.0 m, 3.0 m) at t _ 0; (6.0 m, 7.0 m) at t _ 2.0 s; and (13 m, 14 m) at t _ 5.0 s. (a) Find the magnitude of the average velocity from t _ 0 to t _ 2.0 s. (b) Find the magnitude of the average velocity from t _ 0 to t _ 5.0 s.

A particle moving at a velocity of 4.0 m/s in the _x direction is given an acceleration of 3.0 m/s2 in the _y direction for 2.0 s. Find the final speed of the particle.

Initially, a swift-moving hawk is moving due west with a speed of 30 m/s; 5.0 s later it is moving due north with a speed of 20 m/s. (a) What are the magnitude and direction of during this 5.0-s interval? (b) What are the magnitude and direction of during this 5.0-s interval?

At t _ 0, a particle located at the origin has a velocity of 40 m/s at _ _ 45. At t _ 3.0 s, the particle is at x _ 100 m and y _ 80 m and has a velocity of 30 m/s at _ _ 50. Calculate (a) the average velocity, and (b) the average acceleration of the particle during this 3.0-s interval.

At time zero, a particle is at x _ 4.0 m and y _ 3.0 m and has velocity . The acceleration of the particle is constant and is given by (4.0 m/s2) _ (3.0 m/s2) . (a) Find the velocity at t _ 2.0 s. (b) Express the position vector at t _ 4.0 s in terms of . In addition, give the magnitude and direction of the position vector at this time.

Aparticle has a position vector given by _ (30t) _ (40t _ 5t2) , where r is in meters and t is in seconds. Find the instantaneous-velocity and instantaneous-acceleration vectors as functions of time t.

A particle has a constant acceleration of (6.0 m/s2) _ (4.0 m/s2) . At time t _ 0, the velocity is zero and the position vector is 0 _ (10 m) . (a) Find the velocity and position vectors as functions of time t. (b) Find the equation of the particles path in the xy plane and sketch the path.

Starting from rest at a dock, a motor boat on a lake heads north while gaining speed at a constant 3.0 m/s2 for 20 s. The boat then heads west and continues for 10 s at the speed that it had at 20 s. (a) What is the average velocity of the boat during the 30-s trip? (b) What is the average acceleration of the boat during the 30-s trip? (c) What is the displacement of the boat during the 30-s trip?

Starting from rest at point A, you ride your motorcycle north to point B 75.0 m away, increasing speed at a steady rate of 2.00 m/s2. You then gradually turn toward the east along a circular path of radius 50.0 m at constant speed from B to point C, until your direction of motion is due east at C. You then continue eastward, slowing at a steady rate of 1.00 m/s2 until you come to rest at point D. (a) What is your average velocity and acceleration for the trip from A to D? (b) What is your displacement during your trip from A to C? (c) What distance did you travel for the entire trip from A to D? SSM

Aplane flies at an airspeed of 250 km/h. Awind is blowing at 80 km/h toward the direction 60 east of north. (a) In what direction should the plane head in order to fly due north relative to the ground? (b) What is the speed of the plane relative to the ground?

A swimmer heads directly across a river, swimming at 1.6 m/s relative to the water. She arrives at a point 40 m downstream from the point directly across the river, which is 80 m wide. (a) What is the speed of the river current? (b) What is the swimmers speed relative to the shore? (c) In what direction should the swimmer head in order to arrive at the point directly opposite her starting point?

A small plane departs from point A heading for an airport 520 km due north at point B. The airspeed of the plane is 240 km/h and there is a steady wind of 50 km/h blowing directly toward the southeast. Determine the proper heading for the plane and the time of flight.

Two boat landings are 2.0 km apart on the same bank of a stream that flows at 1.4 km/h. A motorboat makes the round trip between the two landings in 50 min. What is the speed of the boat relative to the water?

ENGINEERING APPLICATION, CONTEXT-RICH During a radio-controlled model-airplane competition, each plane must fly from the center of a 1.0-km-radius circle to any point on the circle and back to the center. The winner is the plane that has the shortest round-trip time. The contestants are free to fly their planes along any route as long as the plane begins at the center, travels to the circle, and then returns to the center. On the day of the race, a steady wind blows out of the north at 5.0 m/s. Your plane can maintain an air speed of 15 m/s. Should you fly your plane upwind on the first leg and downwind on the trip back, or across the wind, flying east and then west? Optimize your chances by calculating the round-trip time for both routes using your knowledge of vectors and relative velocities. With this prerace calculation, you can determine the best route and have a major advantage over the competition!

CONTEXT-RICH You are piloting a small plane that can maintain an air speed of 150 kt (knots, or nautical miles per hour) and you want to fly due north (azimuth _ 000) relative to the ground. (a) If a wind of 30 kt is blowing from the east (azimuth _ 090), calculate the heading (azimuth) you must ask your copilot to maintain. (b) At that heading, what will be your ground speed?

Car A is traveling east at 20 m/s toward an intersection. As car A crosses the intersection, car B starts from rest 40 m north of the intersection and moves south steadily gaining speed at 2.0 m/s2. Six seconds after A crosses the intersection find (a) the position of B relative to A, (b) the velocity of B relative to A, (c) the acceleration of B relative to A. Hint: Let the unit vectors and be toward the east and north, respectively, and express your answers using and .

While walking between gates at an airport, you notice a child running along a moving walkway. Estimating that the child runs at a constant speed of 2.5 m/s relative to the surface of the walkway, you decide to try to determine the speed of the walkway itself. You watch the child run on the entire 21-m walkway in one direction, immediately turn around, and run back to his starting point. The entire trip takes a total elapsed time of 22 s. Given this information, what is the speed of the moving walkway relative to the airport terminal?

Ben and Jack are shopping in a department store. Ben leaves Jack at the bottom of the escalator and walks east at a speed of 2.4 . Jack gets on the escalator, which is inclined at an angle of 37 above the horizontal, and travels eastward and upward at a speed of 2.0 . (a) What is the velocity of Ben relative to Jack? (b) At what speed should Jack walk up the escalator so that he is always directly above Ben (until he reaches the top)?

A juggler traveling in a train on level track throws a ball straight up, relative to the train, with a speed of 4.90 m/s. The train has a velocity of 20.0 m/s due east. As observed by the juggler, (a) what is the balls total time of flight, and (b) what is the displacement of the ball during its rise? According to a friend standing on the ground next to the tracks, (c) what is the balls initial speed, (d) what is the angle of the launch, and (e) what is the displacement of the ball during its rise?

What is the magnitude of the acceleration of the tip of the minute hand of the clock in Problem 38? Express it as a fraction of the magnitude of free-fall acceleration g.

CONTEXT-RICH You are designing a centrifuge to spin at a rate of 15,000 rev/min. (a) Calculate the maximum centripetal acceleration that a test-tube sample held in the centrifuge arm 15 cm from the rotation axis must withstand. (b) It takes 1 min, 15 s for the centrifuge to spin up to its maximum rate of revolution from rest. Calculate the magnitude of the tangential acceleration of the centrifuge while it is spinning up, assuming that the tangential acceleration is constant.

Earth rotates on its axis once every 24 hours, so that objects on its surface execute uniform circular motion about the axis with a period of 24 hours. Consider only the effect of this rotation on the person on the surface. (Ignore Earths orbital motion about the Sun.) (a) What is the speed and what is the magnitude of the acceleration of a person standing on the equator? (Express the magnitude of this acceleration as a percentage of g.) (b) What is the direction of the acceleration vector? (c) What is the speed and what is the magnitude of the acceleration of a person standing on the surface at 35N latitude? (d) What is the angle between the direction of the acceleration of the person at 35N and the direction of the acceleration of the person at the equator if both persons are at the same longitude?

Determine the acceleration of the moon toward Earth, using values for its mean distance and orbital period from the Terrestrial and Astronomical Data table in this book. Assume a circular orbit. Express the acceleration as a fraction of g.

(a) What are the period and speed of a person on a carousel if the person has an acceleration with a magnitude of 0.80 m/s2 when she is standing 4.0 m from the axis? (b) What are her acceleration magnitude and speed if she then moves to a distance of 2.0 m from the carousel center and the carousel keeps rotating with the same period?

Pulsars are neutron stars that emit X rays and other radiation in such a way that we on Earth receive pulses of radiation from the pulsars at regular intervals equal to the period that they rotate. Some of these pulsars rotate with periods as short as 1 ms! The Crab Pulsar, located inside the Crab Nebula in the constellation Orion, has a period currently of length 33.085 ms. It is estimated to have an equatorial radius of 15 km, an average radius for a neutron star. (a) What is the value of the centripetal acceleration of an object on the surface at the equator of the pulsar? (b) Many pulsars are observed to have periods that lengthen slightly with time, a phenomenon called spin down. The rate of slowing of the Crab Pulsar is 3.5 _ 10_13 s per second, which implies that if this rate remains constant, the Crab Pulsar will stop spinning in 9.5 _ 1010 s (about 3000 years from today). What is the tangential acceleration of an object on the equator of this neutron star?

BIOLOGICAL APPLICATION Human blood contains plasma, platelets, and blood cells. To separate the plasma from other components, centrifugation is used. Effective centrifugation requires subjecting blood to an acceleration of 2000g or more. In this situation, assume that blood is contained in test tubes that are 15 cm long and are full of blood. These tubes ride in the centrifuge tilted at an angle of 45.0o above the horizontal (Figure 3-34). (a) What is the distance of a sample of blood from the rotation axis of a centrifuge rotating at 3500 rpm, if it has an acceleration of 2000g? (b) If the blood at the center of the tubes revolves around the rotation axis at the radius calculated in Part (a), calculate the accelerations experienced by the blood at each end of the test tube. Express all accelerations as multiples of g.

While trying out for the position of pitcher on your high school baseball team, you throw a fastball at 87 mi/h toward home plate, which is 18.4 m away. How far does the ball drop due to effects of gravity by the time it reaches home plate? (Ignore any effects due to air resistance.)

A projectile is launched with speed v0 at an angle of _0 above the horizontal. Find an expression for the maximum height it reaches above its starting point in terms of v0, _0, and g. (Ignore any effects due to air resistance.)

A cannonball is fired with initial speed v0 at an angle 30 above the horizontal from a height of 40 m above the ground. The projectile strikes the ground with a speed of 1.2v0. Find v0. (Ignore any effects due to air resistance.)

In Figure 3-35, what is the minimum initial speed of the dart if it is to hit the monkey before the monkey hits the ground, which is 11.2 m below the initial position of the monkey, if x _ 50 m and h _ 10 m? (Ignore any effects due to air resistance.)

A projectile is launched from ground level with an initial speed of 53 m/s. Find the launch angle (the angle the initial velocity vector is above the horizontal) such that the maximum height of the projectile is equal to its horizontal range. (Ignore any effects due to air resistance.)

A ball launched from ground level lands 2.44 s later on a level field 40.0 m away from the launch point. Find the magnitude of the initial velocity vector and the angle it is above the horizontal. (Ignore any effects due to air resistance.)

Consider a ball that is launched from ground level with initial speed v0 at an angle _0 above the horizontal. If we consider its speed to be v at some height h above the ground, show that for a given value of h, v is independent of _0. (Ignore any effects due to air resistance.)

At of its maximum height, the speed of a projectile is of its initial speed. What was its launch angle? (Ignore any effects due to air resistance.)

A cargo plane is flying horizontally at an altitude of 12 km with a speed of 900 km/h when a large crate falls out of the rear loading ramp. (Ignore any effects due to air resistance.) (a) How long does it take the crate to hit the ground? (b) How far horizontally is the crate from the point where it fell off when it hits the ground? (c) How far is the crate from the aircraft when the crate hits the ground, assuming that the plane continues to fly with the same velocity?

Wile E. Coyote (Carnivorus hungribilus) is chasing the Roadrunner (Speedibus cantcatchmi) yet again. While running down the road, they come to a deep gorge, 15.0 m straight across and 100 m deep. The Roadrunner launches himself across the gorge at a launch angle of 15 above the horizontal, and lands with 1.5 m to spare. (a) What was the Roadrunners launch speed? (b) Wile E. Coyote launches himself across the gorge with the same initial speed, but at a different launch angle. To his horror, he is short of the other lip by 0.50 m. What was his launch angle? (Assume that it was less than 15.)

A cannon barrel is elevated 45 above the horizontal. It fires a ball with a speed of 300 m/s. (a) What height does the ball reach? (b) How long is the ball in the air? (c) What is the horizontal range of the cannon ball? (Ignore any effects due to air resistance.)

A stone thrown horizontally from the top of a 24-m tower hits the ground at a point 18 m from the base of the tower. (Ignore any effects due to air resistance.) (a) Find the speed with which the stone was thrown. (b) Find the speed of the stone just before it hits the ground.

A projectile is fired into the air from the top of a 200-m cliff above a valley (Figure 3-36). Its initial velocity is 60 m/s at 60 above the horizontal. Where does the projectile land? (Ignore any effects due to air resistance.)

The range of a cannonball fired horizontally from a cliff is equal to the height of the cliff. What is the direction of the velocity vector of the projectile as it strikes the ground? (Ignore any effects due to air resistance.)

An archer fish launches a droplet of water from the surface of a small lake at an angle of 60 above the horizontal. He is aiming at a juicy spider sitting on a leaf 50 cm to the east and on a branch 25 cm above the water surface. The fish is trying to knock the spider into the water so that the fish can eat the spider. (a) What must the speed of the water droplet be for the fish to be successful? (b) When it hits the spider, is the droplet rising or falling?

CONTEXT-RICH You are trying out for the position of place-kicker on a professional football team. With the ball teed 50.0 m from the goalposts with a crossbar 3.05 m off the ground, you kick the ball at 25.0 m/s and 30 above the horizontal. (a) Is the field goal attempt good? (b) If so, by how much does it clear the bar? If not, by how much does it go under the bar? (c) How far behind the plane of the goalposts does the ball land?

The speed of an arrow fired from a compound bow is about 45.0 m/s. (a) A Tartar archer sits astride his horse and launches an arrow into the air, elevating the bow at an angle of 10 above the horizontal. If the arrow is 2.25 m above the ground at launch, what is the arrows horizontal range? Assume that the ground is level, and ignore any effects due to air resistance. (b) Now assume that his horse is at full gallop, moving in the same direction that the archer will fire the arrow. Also assume that the archer elevates the bow at the same elevation angle as in Part (a) and fires. If the horses speed is 12.0 m/s, what is the arrows horizontal range now?

The roof of a two-story house makes an angle of 30 with the horizontal. A ball rolling down the roof rolls off the edge at a speed of 5.0 m/s. The distance to the ground from that point is 7.0 m. (a) How long is the ball in the air? (b) How far from the base of the house does it land? (c) What is its speed and direction just before landing?

Compute dR/d_0 from and show that setting dR/d_0 _ 0 gives _0 _ 45 for the maximum range.

In a science fiction short story written in the 1970s, Ben Bova described a conflict between two hypothetical colonies on the moon—one founded by the United States and the other by the USSR. In the story, colonists from each side started firing bullets at each other, only to find to their horror that their rifles had large enough muzzle velocities so that the bullets went into orbit. (a) If the magnitude of free-fall acceleration on the moon is \(1.67~ \mathrm{m/s}^2\), what is the maximum range of a rifle bullet with a muzzle velocity of 900 m/s? (Assume the curvature of the surface of the moon is negligible.) (b) What would the muzzle velocity have to be to send the bullet into a circular orbit just above the surface of the moon?

On a level surface, a ball is launched from ground level at an angle of 55 above the horizontal, with an initial speed of 22 m/s. It lands on a hard surface, and bounces, reaching a peak height of 75% of the height it reached on its first arc. (Ignore any effects due to air resistance.) (a) What is the maximum height reached in its first parabolic arc? (b) How far horizontally from the launch point did it strike the ground the first time? (c) How far horizontally from the launch point did the ball strike the ground the second time? Assume the horizontal component of the velocity remains constant during the collision of the ball with the ground. Hint: You cannot assume that the angle with which the ball leaves the ground after the first collision is the same as the initial launch angle.

In the text, we calculated the range for a projectile that lands at the same elevation from which it is fired as . A golf ball hit from an elevated tee at 45.0 m/s and an angle of 35.0 lands on a green 20.0 m below the tee (Figure 3-37). (Ignore any effects due to air resistance.) (a) Calculate the range using the equation even though the ball is hit from an elevated tee. (b) Show that the range for the more general problem (Figure 3-37) is given by where y is the height of the green above the tee. That is, y__h. (c) Compute the range using this formula. What is the percentage error in neglecting the elevation of the green?

MULTISTEP In the text, we calculated the range for a projectile that lands at the same elevation from which it is fired as if the effects of the air resistance are negligible. (a) Show that for the same conditions the change in the range for a small change in free-fall acceleration g, and the same initial speed and angle, is given by _R/R _ __g/g. (b) What would be the length of a homerun at a high altitude where g is 0.50% less than at sea level if the homerun at sea level traveled 400 ft?

MULTISTEP, APPROXIMATION In the text, we calculated the range for a projectile that lands at the same elevation from which it is fired as if the effects of the air resistance are negligible. (a) Show that for the same conditions the change in the range for a small change in launch speed, and the same initial angle and free-fall acceleration, is given by _R/R _ 2_v0/v0. (b) Suppose a projectiles range was 200 m. Use the formula in Part (a) to estimate its increase in range if the launch speed were increased by 20.0%. (c) Compare your answer in (b) to the increase in range by calculating the increase in range directly from . If the results for Parts (b) and (c) are different, is the estimate too small or large, and why?

Aprojectile, fired with unknown initial velocity, lands 20.0 s later on the side of a hill, 3000 m away horizontally and 450 m vertically above its starting point. (Ignore any effects due to air resistance.) (a) What is the vertical component of its initial velocity? (b) What is the horizontal component of its initial velocity? (c) What was its maximum height above its launch point? (d) As it hit the hill, what speed did it have and what angle did its velocity make with the vertical?

Aprojectile is launched over level ground at an initial elevation angle of _. An observer standing at the launch site sees the projectile at the point of its highest elevation and measures the angle _ shown in Figure 3-38. Show that tan_ _ . (Ignore any effects due to air resistance.) SSM

A toy cannon is placed on a ramp that has a slope of angle _. (a) If the cannonball is projected up the hill at an angle of _0 above the horizontal (Figure 3-39) and has a muzzle speed of v0, show that the range R of the cannonball (as measured along the ramp) is given by Ignore any effects due to air resistance.

A rock is thrown from the top of a 20-m-high building at an angle of 53 above the horizontal. (a) If the horizontal range of the throw is equal to the height of the building, with what speed was the rock thrown? (b) How long is it in the air? (c) What is the velocity of the rock just before it strikes the ground? (Ignore any effects due to air resistance.)

A woman throws a ball at a vertical wall 4.0 m away (Figure 3-40). The ball is 2.0 m above ground when it leaves the womans hand with an initial velocity of 14 m/s at 45, as shown. When the ball hits the wall, the horizontal component of its velocity is reversed; the vertical component remains unchanged. (a) Where does the ball hit the ground? (b) How long was the ball in the air before it hit the wall? (c) Where did the ball hit the wall? (d) How long was the ball in the air after it left the wall? Ignore any effects due to air resistance.

ENGINEERING APPLICATION Catapults date from thousands of years ago, and were used historically to launch everything from stones to horses. During a battle in what is now Bavaria, inventive artillerymen from the united German clans launched giant spaetzle from their catapults toward a Roman fortification whose walls were 8.50 m high. The catapults launched the spaetzle projectiles from a height of 4.00 m above the ground and a distance of 38.0 m from the walls of the fortification at an angle of 60.0 degrees above the horizontal (Figure 3-41). If the projectiles were to hit the top of the wall, splattering the Roman soldiers atop the wall with pulverized pasta, (a) what launch speed was necessary? (b) How long were the spaetzle in the air? (c) At what speed did the projectiles hit the wall? Ignore any effects due to air resistance.

The distance from the pitchers mound to home plate is 18.4 m. The mound is 0.20 m above the level of the field. A pitcher throws a fastball with an initial speed of 37.5 m/s. At the moment the ball leaves the pitchers hand, it is 2.30 m above the mound. (a) What should the angle between and the horizontal be so that the ball crosses the plate 0.70 m above ground? (Ignore any effects due to air resistance.) (b) With what speed does the ball cross the plate?

You are watching your friend play hockey. During the course of the game, he strikes the puck in such a way that, when it is at its highest point, it just clears the surrounding 2.80-m-high Plexiglas wall that is 12.0 m away. Find (a) the vertical component of its initial velocity, (b) the time it takes to reach the wall, and (c) the horizontal component of its initial velocity, and its initial speed and angle. (Ignore any effects due to air resistance.)

Carlos is on his trail bike, approaching a creek bed that is 7.0 m wide. A ramp with an incline of 10 has been built for daring people who try to jump the creek. Carlos is traveling at his bikes maximum speed, 40 km/h. (a) Should Carlos attempt the jump or brake hard? (b) What is the minimum speed a bike must have to make this jump? Assume equal elevations on either side of the creek. (Ignore any effects due to air resistance.)

If a bullet that leaves the muzzle of a gun at 250 m/s is to hit a target 100 m away at the level of the muzzle (1.7 m above the level ground), the gun must be aimed at a point above the target. (a) How far above the target is that point? (b) How far behind the target will the bullet strike the ground? (Ignore any effects due to air resistance.)

During a do-it-yourself roof repair project, you are on the roof of your house and accidentally drop your hammer. The hammer then slides down the roof at constant speed of 4.0 m/s. The roof makes an angle of 30 with the horizontal, and its lowest point is 10 m from the ground. (a) How long after leaving the roof does the hammer hit the ground? (b) What is the horizontal distance traveled by the hammer between the instant it leaves the roof and the instant it hits the ground? (Ignore any effects due to air resistance.)

A squash ball typically rebounds from a surface with 25% of the speed with which it initially struck the surface. Suppose a squash ball is served in a shallow trajectory, from a height above the ground of 45 cm, at a launch angle of 6.0 degrees above the horizontal, and at a distance of 12 m from the front wall. (a) If it strikes the front wall exactly at the top of its parabolic trajectory, determine how high above the floor the ball strikes the wall. (b) How far horizontally from the wall does it strike the floor, after rebounding? (Ignore any effects due to air resistance.)

A football quarterback throws a pass at an angle of 36.5. He releases the pass 3.50 m behind the line of scrimmage. His receiver left the line of scrimmage 2.50 s earlier, goes straight downfield at a constant speed of 7.50 m/s. In order that the pass land gently in the receivers hands without the receiver breaking stride, with what speed must the quarterback throw the pass? Assume that the ball is released at the same height it is caught and that the receiver is straight downfield from the quarterback at the time of release. (Ignore any effects due to air resistance.)

Suppose a test pilot is able to safely withstand an acceleration of up to 5.0 times the acceleration due to gravity (that is, remain conscious and alert enough to fly). During the course of maneuvers, he is required to fly the plane in a horizontal circle at its top speed of 1900 mi/h. (a) What is the radius of the smallest circle he will be able to safely fly the plane in? (b) How long does it take him to go halfway around this minimum-radius circle?

Aparticle moves in the xy plane with constant acceleration. At t _ 0 the particle is at _ (4.0 m) _ (3.0 m) , with velocity . At t _ 2.0 s, the particle has moved to _ (10 m) _ (2.0 m) and its velocity has changed to _ (5.0 m/s) _ (6.0 m/s) . (a) Find . (b) What is the acceleration of the particle? (c) What is the velocity of the particle as a function of time? (d) What is the position vector of the particle as a function of time?

Plane A is flying due east at an air speed of 400 mph. Directly below, at a distance of 4000 ft, plane B is headed due north, flying at an air speed of 700 mph. Find the velocity vector of plane B relative to A.

A diver steps off the cliffs at Acapulco, Mexico, 30.0 m above the surface of the water. At that moment, he activates his rocket-powered backpack horizontally, which gives him a constant horizontal acceleration of 5.00 m/s2 but does not affect his vertical motion. (a) How long does he take to reach the surface of the water? (b) How far out from the base of the cliff does he enter the water, assuming the cliff is vertical? (c) Show that his flight path is a straight line. (Ignore any effects of air resistance.)

Asmall steel ball is projected horizontally off the top of a long flight of stairs. The initial speed of the ball is 3.0 m/s. Each step is 0.18 m high and 0.30 m wide. Which step does the ball strike first?

Suppose you can throw a ball a maximum horizontal distance L when standing on level ground. How far can you throw it from the top of a building of height h if you throw it at (a) \(0^\circ\), (b) \(30^\circ\), (c) \(45^\circ\)? (Ignore any effects due to air resistance.)

Darlene is a stunt motorcyclist in a traveling circus. For the climax of her show, she takes off from the ramp at angle _0, clears a ditch of width L, and lands on an elevated ramp (height h) on the other side (Figure 3-42). (a) For a given height h, find the minimum necessary takeoff speed vmin needed to make the jump successfully. (b) What is vmin for L _ 8.0 m, _ _ 30, and h _ 4.0 m? (c) Show that regardless of her takeoff speed, the maximum height of the platform is h Ltan_0. Interpret this result physically. (Neglect any effects due to air resistance and treat the rider and the bike as if they were a single particle.)

Asmall boat is headed for a harbor 32 km directly northwest of its current position when it is suddenly engulfed in heavy fog. The captain maintains a compass bearing of northwest and a speed of 10 km/h relative to the water. The fog lifts 3.0 h later and the captain notes that he is now exactly 4.0 km south of the harbor. (a) What was the average velocity of the current during those 3.0 h? (b) In what direction should the boat have been heading to reach its destination along a straight course? (c) What would its travel time have been if it had followed a straight course?

Galileo showed that, if any effects due to air resistance are ignored, the ranges for projectiles (on a level field) whose angles of projection exceed or fall short of \(45^\circ\) by the same amount are equal. Prove Galileo’s result.

Two balls are thrown with equal speeds from the top of a cliff of height h. One ball is thrown at an angle of _ above the horizontal. The other ball is thrown at an angle of _ below the horizontal. Show that each ball strikes the ground with the same speed, and find that speed in terms of h and the initial speed v0. (Ignore any effects due to air resistance.)

In his car, a driver tosses an egg vertically from chest height so that the peak of its path is just below the ceiling of the passenger compartment, which is 65 cm above his release point. He catches the egg at the same height that he releases it. If you are a roadside observer and measure the horizontal distance between catch and release points to be 19 m, (a) how fast is the car moving? (b) In your reference frame, at what angle above the horizontal was the egg thrown? SSM

A straight line is drawn on the surface of a 16-cm-radius turntable from the center to the perimeter. A bug crawls along this line from the center outward as the turntable spins counterclockwise at a constant 45 rpm. Its walking speed relative to the turntable is a steady 3.5 cm/s. Let its initial heading be in the positive x direction. As the bug reaches the edge of the turntable (still traveling at 3.5 cm/s radially, relative to the turntable), what are the x and y components of the velocity of the bug?

On a windless day, a stunt pilot is flying his vintage World War I Sopwith Camel from Dubuque, Iowa, to Chicago, Illinois, for an air show. Unfortunately, he is unaware that his planes ancient magnetic compass has a serious problem in that what it records as north is in fact 16.5 east of true north. At one moment during his flight, the airport in Chicago notifies him that he is, in reality, 150 km due west of the airport. He then turns due east, according to his planes compass, and flies for 45 minutes at 150 km/h. At that point, he expects to see the airport and begin final descent. What is the planes actual distance from Chicago and what should be the pilots heading if he is to fly directly toward Chicago?

ENGINEERING APPLICATION, CONTEXT-RICH A cargo plane in flight lost a package because somebody forgot to close the rear cargo doors. You are on the team of safety experts trying to analyze what happened. From the point of takeoff, while climbing to altitude, the airplane traveled in a straight line and at a constant speed of 275 mi/h at an angle of 37 above the horizontal. During the ascent, the package slid off the back ramp. You found the package in a field a distance of 7.5 km from the takeoff point. To complete the investigation you need to know exactly how long after takeoff the package slid off the back ramp of the plane. (Consider the sliding speed to be negligible.) Calculate the time at which the package fell off the back ramp. (Ignore any effects due to air resistance.)