Projectile motion practice problems. Physics APB Mr. Szakiel 10-1-12
1. A tiger leaps horizontally from a 7.5 m high rock with a speed of 4.5 m/s. How far from the base of the rock will she land? 2. A diver running 1.6 m/s dives out horizontally from the edge of a vertical cliff and reaches the water below 3.0 s later. How high was the cliff and how far from its base did the diver hit the water?
3. A fire hose held near the ground shoots water at a speed of 6.5 m/s. At what angle(s) should the nozzle point in order that the water land 2.0 m away? Why are there two different angles? 4.
Romeo is chucking pebbles gently up to Juliet's window, and he w ants the pebbles to hit the window with only a horizontal component of velocity. He is standing at the edge of a rose garden 8.0 m below her window and 9 .0 m from the base of the wall. How fast are the pebbles going when they hit her window?
5. A ball is thrown horizontally from the roof of a building 56 m tall and lands 45 m from the base. What was the ball's initial speed? 6. Show that the speed with which a projectile leaves the ground is e qual to its speed just before it strikes the ground at the end of its journey, assuming the firing level equals the landing level. 7. A football is kicked at ground level with a speed of 20.0 m/s at an angle of 37.0° to the horizontal. How much later does it hit the ground? 8. A ball thrown horizontally at 22.2 m/s from the roof of a building lands 36.0 m from the base of the building. How high is the building? 9. A shot putter throws the shot with an initial speed of 14 m/s at a 40° angle to the horizontal. Calculate the horizontal distance traveled by the shot if it leaves the athlete's hand at a height of 2.2 m above the ground. 10. Determine how much farther a person can jump on the Moon as compared to the Earth if the takeoff speed and angle are the same. The acceleration due to gravity on the Moon is one sixth what it is on Earth. 11. An athlete executing a long jump leaves the ground at a 30° angle and travels 7.80 m. (a) What was the takeoff speed? (b) If this speed were increased by just 5.0 percent, how much longer would the jump be? 12. The pilot of an airplane traveling 160km/h wants to drop supplies to flood victims isolated on a patch of land 160 m below. The supplies should be dropped how many seconds before the plane is directly overhead? 13. A hunter aims directly at a target (on the same level) 120 m away, (a) If the bullet leaves the gun at a speed of 250 m/s. by how much will it miss the target? (b) At what angle should the gun be aimed so the target will be hit? 14. Show that the time required for a projectile to reach its highest point is equal to the time for it to return from this highest point to its original height. 15. A projectile is fired with an initial speed of 40.0 m/s. Plot on graph paper its trajectory for initial projection angles of θ = 15°. 30°, 45°, 60°. 75°, and 90°. Plot at least 10 points for each curve.
16. A projectile is fired with an initial speed of 75.2 m/s at an angle of 34.5° above the horizontal on a long flat firing range. Determine (a) the maximum height reached by the projectile, (b) the total time in the air, (c) the total horizontal distance covered (that is, the range), and (d) the velocity of the projectile 1.50 s after firing. ______________________________________________________________________________ 17. A meteor streaking through the night sky is located with radar. At point A its coordinates are (5.00 km, 1.20 km), and 14 s later it has moved to point B with coordinates (6.24 km, 0.925 km). Find (a) the x and y components of its average velocity between A and B and (b) the magnitude and direction of its average velocity between these t wo points. 18. At an air show, a jet plane has velocity components vx = 625 km/h and vy = 415 km/h at time 3.85 s and vx = 838 km/h and vy = 365 km/h at time 6.52 s. For this time interval, find (a) the x and y components of the plane's average acceleration and (b) the magnitude and direction of its average acceleration.
19. A dragonfly flies from point A to point B along the path shown above in 1.50 s. (a) Find the x and y components of its position vector at point A. (b) What are the magnitude and direction of its position vector at A? (c) Find the x and y components of the dragonfly's average velocity between A and B. (d) What are the magnitude and direction of its average velocity between these two points? 20. In the previous problem, if the dragonfly's velocity components at point A are vx = 6.20 m/s and Vy =5.11 m/s and at point B are vx = 4.05 m/s and vy = 8.26 m/s, find (a) the x and y components of the dragonfly's average acceleration between A and B and (b) the magnitude and direction of its average acceleration between those points.
21. An athlete starts at point A and runs at a constant speed of 6.0 m/s around a round track 100 m in diameter, as shown above. Find the x and y components of this runner's average velocity and average acceleration between points (a) A and B, (b) A and C, (c) C and D, and (d) A and A (a full lap), (e) Calculate the magnitude of the runner's average velocity between A and B. Is his average speed equal to the magnitude of his average velocity? Why or why not? (f) How can his velocity be changing if he is running at constant speed? 22. A stone is thrown horizontally at 30.0 m/s from the top of a very tall cliff. (a) Calculate its horizontal position and vertical position at 2 s intervals for the first 10.0 s. (b) Plot your positions from part (a) to scale. Then connect your points with a smooth curve to show the trajectory of the stone. 23. A marksman fires a .22-caliber rifle horizontally at a target; the bullet has a muzzle velocity with magnitude 750 ft/s. How much does the bullet drop i n flight if the target is (a) 50.0 yds. away and (b) 150.0 yds. away? 24. A physics book slides off a horizontal tabletop with a speed of 1.10 m/s. It strikes the floor in 0.350 s. Ignore air resistance. Find (a) the height of the tabletop above the floor, (b) the horizontal distance from the edge of the table to the point where the book strikes the floor, and (c) the horizontal and vertical components of the book's velocity, and the magnitude and direction of its velocity, just before the book reaches the floor. 25. A tennis ball rolls off the edge of a tabletop 0.750 m above the floor and strikes the floor at a point 1.40 m horizontally from the edge of the table, (a) Find the time of flight of the ball, (b) Find the magnitude of the initial velocity of the ball, (c) Find the magnitude and direction of the velocity of the ball just before it strikes the floor. 26. A military helicopter on a training mission is flying horizontally at a speed of 60 0 m/s when it accidentally drops a bomb (fortunately, not armed) at an elevation of 300 m. You can ignore air resistance, (a) How much time is required for the bomb to reach the earth? (b) How far does it travel horizontally while falling? (c) Find the horizontal and vertical components of the bomb's velocity just before it strikes the earth, (d) Draw graphs of the horizontal distance vs. time and the vertical distance vs. time for the bomb's motion, (e) If the velocity of the helicopter remains constant, where is the helicopter when the bomb hits the ground? 27. Two crickets, Chirpy and Milada, jump from the top of a vertical cliff. Chirpy jumps horizontally at 95.0 cm/s, while Milada simply drops and reaches the ground in 3.50 s. How far from the base of the cliff will Chirpy hit the ground?
28. A daring 510 N swimmer dives off a cliff with a running horizontal leap, as shown above. 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? 29. Leaping the river. A 10,000 N car comes to a bridge during a storm and finds the bridge washed out. The 650 N driver must get to the other side, so he decides to try leaping it with his car. The side the car is on is 21.3 m above the river, while the opposite side is a mere 1.80 m above the river. The river itself is a raging torrent 61.0 m wide, (a) How fast should the car be traveling just as it leaves the cliff in order to clear the river and land safely on the opposite side? (b) What is the speed of the car just before it lands safely on the other side? 30. A football is thrown with an initial upward velocity component of 15.0 m/s and a horizontal velocity component of 18.0 m/s. (a) How much time is required for the football to reach the highest point in its trajectory? (b) How high does it get above its release point? (c) How much time after it is thrown does it take to return to its original height? How does this time compare with what you calculated in part (b)? Is your answer reasonable? (d) How far has the football traveled horizontally from its original position? 31. A tennis player hits a ball at ground level, giving it an initial velocity of 24 m/s at 57° above the horizontal, (a) What are the horizontal and vertical components of the ball's initial velocity? (b) How high above the ground does the ball go? (c) How long does it take the ball to reach its maximum height? (d) What are the ball's velocity and acceleration at its highest point? (e) For how long a time is the ball in the air? (f) When this ball lands on the court, how far is it from the place where it was hit? 32. A baseball is hit by a batter at 51.0° above the horizontal with an initial speed of 65.2 m/s. What are the horizontal and vertical components of its initial velocity? Sketch this vector and its components to scale, (b) In a skeet shoot, a bullet is shot upward with an initial vertical velocity component of 818 m/s and an initial horizontal component of 511
m/s. Find the initial speed of the bullet and the angle it makes above the horizontal? Sketch this vector and its components to scale. 33. A major leaguer hits a baseball so that it leaves the bat at a speed of 30.0 m/s and at an angle of 36.9° above the horizontal. You can ignore air resistance, (a) At what two times is the baseball at a height of 10.0 m above the point at which it left the bat? (b) Calculate the horizontal and vertical components of the baseball's velocity at each of the two times you found in part (a), (c) What are the magnitude and direction of the baseball's velocity when it returns to the level at which it left the bat?
34. A balloon carrying a basket is descending at a constant velocity of 20.0 m/s. A person in the basket throws a stone with an initial velocity of 15.0 m/s horizontally perpendicular to the path of the descending balloon, and 4.00 s later this person sees the rock strike the ground. (See Figure above.) (a) How high was the balloon when the rock was thrown out? (b) How far horizontally does the rock travel before it hits the ground? (c) At the instant the rock hits the ground, how far is it from the basket? 35. A batted baseball leaves the bat at an angle of 30.0° above the horizontal and is caught by an outfielder 375 ft from home plate at the same height from which it left the bat. (a) What was the initial speed of the ball? (b) How high does the ball rise above the point where it struck the bat? 36. A man stands on the roof of a 15.0 m tall building and throws a rock with a velocity of magnitude 30.0 m/s at an angle of 33.0° above the horizontal. You can ignore air resistance. Calculate (a) the maximum height above the roof reached by the rock, (b) the magnitude of the velocity of the rock just before it strikes the ground, and (c) the horizontal distance from the base of the building to the point where the rock strikes the ground. 37. 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, 31 July 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?
38. A grasshopper leaps into the air from the edge of a vertical cliff, as shown in Figure above. Use information from the figure to fi nd (a) the initial speed of the grasshopper and (b) the height of the cliff. 39. Firemen are shooting stream of water at a burning building. A high pressure hose shoots out the water: with a speed of 25.0 m/s as it leaves the hose nozzle. Once it leaves the hose, the water moves in projectile motion. The firemen adjust the angle of elevation of the hose until the water takes 3.00 s to reach a building 45.0 m away. You can i gnore air resistance; assume that the end of the hose is at ground level, (a) Find the angle of elevation of the hose, (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? 40. Show that a projectile achieves its maximum range when it i s fired at 45° above the horizontal if y = y0. 41. A certain peewee soccer player can kick the ball with a maximum speed of 15.0 m/s. (a) At what angle should he kick it to achieve the maximum range? (b) What is the maximum range of the ball at that angle? (c) How high will the ball go? 42. A certain cannon with a fixed angle of projection has a ran ge of 1500 m. What will be its range if you add more powder so that the initial speed of the cannonball is tripled? 43. A cannon can be adjusted to various angles of elevation to launch shells at a target 1250 m away on level ground. What is the minimum muzzle velocity needed to reach this target? At what angle should the cannon launch the shells?
44. Two archers shoot arrows in the same direction from the same place with the same initial speeds but at different angles. One shoots at 45° above the horizontal, while the other shoots at 60.0°. If the arrow launched at 45° lands 225 m from the archer, how far apart are the two arrows when they land? (You can assume that the arrows start at essentially ground level.) 45. If a certain cannon can shoot a projectile a di stance of 275 m when it is aimed at 66.0° above the horizontal, what is the maximum range the cannon could achieve with the same projectile, and at what angle should it be aimed to do this? 46. Martian Olympics. The world record for the discus throw is 74.08 m, set by Jurgen Schult in 1986. If he had been competing not on Earth, but on Mars, where the acceleration due to gravity is 0.379 what it is on Earth, and if he had thrown the discus in exactly the same way as on earth, what would be his Martian record for this throw? Assume that the discus is released essentially at ground level. 47. 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 55° above the horizontal. How far in f ront of the cattle should the pilot release the hay so that the bales will land at the point where the cattle are stranded?
48. 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 at 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?
49. Don't do this! A girl throws a water filled balloon at an angle of 50.0° above the horizontal with a speed of 12.0 m/s. The horizontal component of the balloon's velocity is directed toward a car that is approaching the girl at a constant speed of 8.00 m/s. (See Figure above.) If the balloon is to hit the car at the same height at which it leaves her hand, what is the maximum distance the car can be from the girl when the balloon is thrown? You can ignore air resistance. 50. The longest home run. According to the Guinness Book of World Records, the longest home run ever measured was hit by Roy "Dizzy" Carlyle in a minor league game. The ball traveled 188 m (618 ft.) before landing on the ground outside the ballpark. (a) Assuming that the ball's initial velocity was 45° above the horizontal, and ignoring air resistance, what did the initial speed of the ball need to be to produce such a home run if the ball was hit at a point 0.9 m (3.0 ft.) above ground level? Assume that the ground was perfectly flat. (b) How far would the ball be above a fence 3.0 m (10 ft.) in height if the fence were 116 m (380 ft.) from home plate? 51. Your spaceship lands on an unknown planet. To determine the local value of g, you ask a steel toed crew member to kick a stone, and you find that if she kicks it at 17.6 m/s at various angles, the maximum range she can achieve is 33.8 m. (a) What is g on this planet? (b) How long was the stone in the air? (c) How high above the ground did the stone go? 52. A current of a river flows steadily from west to east at 1.00 m/s. A boat in the river travels at 3.00 m/s relative to the water. The river is 20.0 m wide with parallel banks. The boat leaves the shore at point A, and point B is directly across the river from that point, (a) At what angle with respect to the north-south direction should the boat head to go directly to point S? (b) If the boat heads as in part (a), how long will it take it to reach point B? (c) Suppose the boat's motor is low on gas, so that the person guiding it wants to get to the opposite bank as quickly as possible and does not care where the boat lands. In what direction should this person steer the boat, and where will it land relative to B?
53. 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 initial velocity of the baseball (the velocity with which the baseball is thrown). (b) Find the magnitude and direction of the velocity of the baseball just before it strikes the building. 54. 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 i s 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? 55. Suppose the boy in the previous problem throws the ball upward at 60.0° above the horizontal, but all else is the same. Repeat parts (a) and (b) of that problem. 56. 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.) 57. A gun shoots a shell into the air with an initial velocity of 100.0 m/s 60.0° above the horizontal on level ground. Sketch quantitative graphs of the shell's horizontal and vertical velocity components as functions of time for the complete motion.
58. Look out! A snowball rolls off a barn roof that slopes downward at an angle of 40.0°. 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. How far from the edge of the barn does the snowball strike the ground if it doesn't strike anything else while falling?
59. Record long jump. The world's record for the women's l ong jump (formerly called the broad jump) is 7.52 m, set in 1988 by Galina Chistyakova. Assuming that she jumped at the angle for the maximum range, what was her takeoff speed in m/s and mph?
60. During a testing program, the path of a projectile is recorded, as shown on Figure above. Use the information in the figure to find (a) the initial speed of the projectile, (b) the angle at which the projectile is fired, and (c) the time during which this projectile is in the air. 61. A world record. In the shot put, a standard track and field event, a 7.3 kg object (the shot) is thrown by releasing it at approximately 40° over a straight left leg. The world record for distance, set by Randy Barnes in 1990, is 23.11 m. Assuming that Barnes released the shot put at 40.0° from a height of 2.00 m above the ground, with what speed, in m/s and mph, did he release it?
62. A Ferris wheel with radius 14.0 m is turning about a horizontal axis through its center, as shown in Figure above. The linear speed of a passenger on the rim is constant and equal to 7.00 m/s. What are the magnitude and direction of the passenger's acceleration as she passes through (a) the lowest point in her circular motion and (b) the highest point in her circular motion? (c) How much time does it take the Ferris wheel to make one revolution?
63. Leaping the river. A physics professor did daredevil stunts in his spare time. His last stunt was an attempt to jump across a river on a motorcycle. (See Figure above.) 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. You can ignore air resistance. (a) What should his speed have been at the top of the ramp for him to have just made it to the edge of the far bank? (b) If his speed was only half the value found in (a), where did he land? 64. In an action adventure film, the hero is supposed to throw a grenade from his car, which is going 90 km/h, to his enemy's car, which is going 110 km/h. The enemy's car i s 15.8 m in front of the hero's when he lets go of the grenade. If the hero throws the grenade so that its initial velocity relative to him is at an angle of 45° above the horizontal, what should the magnitude of the initial velocity be? The cars are traveling in the same direction on a level road. You can ignore air resistance. Find the magnitude of the velocity relative to the hero and relative to the Earth.
65. A 76.0 kg boulder is rolling horizontally at the top of a vertical cliff that is 20.0 m above the surface of a lake, as shown in Figure above. The top of the vertical face of a dam is located 100.0 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 25.0 m below the top of the dam. (a) What must the minimum speed of the rock be just as it leaves the cliff so that it will travel to the plain without striking the dam? (b) How far from the foot of the dam does the rock hit the plain? 66. A 0.17 kg ball is thrown upward with an initial speed of 20.0 m/s f rom the edge of a 45.0 m high cliff. At the instant the ball is thrown, a woman starts running away from the base of the cliff with a constant speed of 6.00 m/s. The woman runs in a straight line on level ground, and air resistance acting on the ball can be ignored. At what angle above the horizontal should the ball be thrown so that the runner will catch it just before it hits the ground? 67. A shell is launched at 150 m/s 53° above the horizontal. When it has reached its highest point, it launches a projectile at a velocity of 100.0 m/s 30.0° above the horizontal relative to the shell. Find (a) the maximum height about the ground that the projectile reaches and (b) its distance from the place where the shell was fired to its landing place when it eventually falls back to the ground.
Answers.
#28
Vo = 1.29 m/s #34
a) ∆y = 158 b) ∆x = 60.0 m c) 98.4 m #38
a) 1.50 m/s b) 4.66 m #48
a) ∆y = 81.6 m b) ∆x = 245 m c) Lands in the cart. #49
∆x
= 29.5 m
#58
6.91 m #60
a) Vo = 15.9 m/s o b) θ = 38.1 c) t = 2.00 s #63
a) Vo = 17.8 m/s b) Vo = 8.9 m/s c) ∆x = 28.4 m #65
a) Vo = 49.0 m/s b) 48 m beyond the foot of the dam. #67
a) 863 m b) 4360 m
