During the 16th century, Galileo Galelei did a series of experiments along with some good, sound reasoning. He concluded that all objects, no matter what their weight, would fall at the same rate if friction with the air could be eliminated.His reasoning was this: Two identical spheres are dropped simultaneously. Everyone will agree they must fall at the same rate. We bring the spheres closer together on successive drops until they are touching as they fall. Again, everyone will agree they still fall at the same rate. Finally, we weld the two together and they will fall at the same rate they did before they were joined. Thus the welded object, with twice the weight of a single sphere will fall no faster.
Although Galileo had crude timing devices and could not accurately measure the acceleration of a free-falling object, he could measure the rate at which spheres rolled down inclined planes. Mathematically extrapolating to a 90° incline, he calculated the free fall acceleration without actually measuring it directly. Although we take this approach to science as routine, in Galileo's time it was a leap toward abstract generalization that was truly revolutionary.
The acceleration of a free falling object at sea level on Earth is 9.8 m/sec2.
1. How far will a rock fall in 6 seconds if it is released from rest?
2. A ball is thrown down at an initial velocity of 18 m/sec. How far will it fall in the first 4 seconds?
3. A ball is thrown up at 22 m/sec. How far will it be above the ground in 2 seconds? ...in 4 seconds?
Notice that there are two terms working in opposition to each other. The v0t term increases steadily while the 1/2 at2 term gets increasingly larger in a negative direction. Thus the ball rises at first, then falls back.
4. Freddy Frog takes a swan dive from a bridge which is 53 meters above the water. Freddy is an expert and can streamline himself to eliminate wind resistance. How long will it take him to reach his aquatic homeland?
5. Often it is necessary to work in units of centimeters or feet, so you need to
know gravity in these units.
a) Convert 9.8 m/s2 to cm/s2.
b) Convert 9.8 m/s2 to ft/s2.
6. Expressed in feet, how far will a rock fall if it drops for 8.2 seconds?
7. How long does it take a needle to drop 2.6 cm?
8**. An inquisitive spelunker comes upon a deep cavern with a vertical drop. It is too deep to see below, so the spelunker drops a rock into the void. 3.8 seconds later the splash is heard. If the speed of the returning sound is 330 m/sec, how deep is the cavern?
As we saw in the previous chapter, when two dimensions are involved we may treat each dimension individually and independently. Thus a falling object will accelerate at 9.8 m/s2 whether it is released from rest or thrown horizontally. In both cases the initial vertical velocity is zero, so at each moment after release they will have fallen an identical vertical distance.
9. A rock is thrown horizontally at 18 m/sec. Where is it 1.5 seconds later?
10. A rock is thrown horizontally at 5.7 m/sec. Find its speed and the direction of its motion 3.6 seconds after release.
11. Rhonda Mulberry Bush kicks a can of Cheese Whiz horizontally at 4.6 m/sec off
a cliff 12.4 m high.
a) How long will it take to land?
b) How far from the cliff will it land?
12. A man fires a bullet horizontally from 1.8 m above the ground. If the bullet's speed is 415 m/sec, how far away will it land?
13. A marble, rolling off the edge of a table is observed to hit the floor 0.77 m from the table. If the table top is 0.86 m high, how fast was the marble traveling when it left the table?
14. A colony of troglodytes has been in a lengthy feud with its neighbors on the adjacent cliff. Colony A finally develops an important military breakthrough: it rolls bombs off its cliff at known rates of speed, thus gaining pinpoint accuracy in its attacks. If the cliffs are separated by 42 m and a bomb is rolled at 6.0 m/sec, how far down the cliff will it land?
15. The troglodyte war continues, and a particularly offensive member of colony B is located 110 m below the top. At what speed must a bomb be rolled to get him?
16. Freddy Frog returns for another diving competition. This time, from the 13 m board he takes off at 6.5 m/sec. If his initial velocity is horizontal, at what angle and at what total speed does Freddy hit the water?
17. Ford tests a new safety system by driving a car off a high cliff. If the vehicle leaves the edge with a speed of 68 ft/sec, what will be its speed and direction in 1.5 seconds?
18*. A family of trolls communicates with another family at the bottom of a two-tiered
cliff as shown. The trolls tie messages to rocks and roll them off the top cliff at the
minimum speed necessary to just miss the 2nd cliff and land below.
a) What is the velocity chosen?
b) How far from the bottom cliff will the projectile land?
19*. This time the gregarious troll family wishes to fire off a message to neighbors who live under the overhang of a cliff. With what velocity must they roll the rock, and how far under the overhang will it hit?
20**. Find the general equation for problem 18, using the parameters shown at right.
So far we have considered projectile problems where the initial velocity has always been horizontal. This has been to simplify the mathematics while instilling the concept that two independent motions are taking place simultaneously. This same concept will apply to motion which begins at some angle other than horizontal. Before studying general projectile motion, however, we must review some of our previous work.
21. Rex Things drops a rock from a 42 foot high cliff. How long does it take to reach the ground?
22. A bolt drops out of a bridge. How fast is it traveling (in m/sec) 2.6 seconds after its release?
23. A soccer ball is kicked vertically up at 22 m/sec. How long will it take to reach the peak of its flight?
24. Foster Pride grabs his Smith and Wessen and fires a bullet vertically up at 1330 ft/sec. How long will it take to reach its highest point? (Neglect friction, of course.)
25. If Priam Raite throws a package of Cheeze-Whiz directly up at 8.6 m/sec, how long will it take to return to ground?
26. A baseball thrown vertically up is aloft 4.8 seconds before returning to Earth. What was its initial velocity in ft/sec?
27. A raindrop, mysteriously falling on the moon, hits at 57.4 m/sec. The moon's gravity is 1.6 m/s2, so how long was the drop falling?
28. Struggling with a fire hose, Jean Zon aims a stream of water vertically up at 17.3 m/sec. How long will it take before the stream is falling at 8.8 m/sec?
29. A Brazilian aborigine fires a dart from his blowgun at 41 m/sec aimed 32° above
the horizontal.
a) What is its initial vertical velocity?
b) How long does it take to return to Earth?
30. A baseball leaves the bat at 76 m/sec, 57° above the horizontal. How long is it in the air?
31. A rock is thrown at 15.2 ft/sec, 41° above the horizon. How long will it be in the air?
32. A Roman centurion fires off a vat of burning pitch from his catapult at 18.2
m/sec, 37° above the horizontal.
a) How long is it in the air?
b) What is the horizontal component of velocity?
c) How far does it get before landing?
33. A bullet is fired at 677 m/sec, 35° above the horizontal. How far does it get before landing? (We neglect air resistance, of course.)
34. A baseball, thrown at 21.7 m/sec 28° above the horizontal will travel how far before hitting the ground?
35. A rock is thrown so that it is in the air 4.3 seconds and lands 36 m from its starting point. What was the initial vertical and horizontal speed?
36. With his trusty slingshot, Morris Cumming fires a pebble which is in the air for 5.3 seconds and lands 73 m from where it took off. What were the initial vertical and horizontal components of velocity?
Look over what we are doing in these solutions. First we split each problem into two separate parts: a vertical one and a horizontal one. In the case of projectile motion, the vertical velocity is the thing that gets the ball into the air. The greater the vertical velocity, the longer the ball will stay aloft.The horizontal velocity has no effect whatsoever on how long the ball stays up, but it's the thing that move the ball forward. We see, then, that the range of a projectile is a combination of the horizontal speed at which the projectile moves toward its target, and the vertical speed which determines how long the projectile is allowed to keep moving.
37. Write the horizontal and vertical equations for each diagram, but do not solve.
38. A B-B takes off at 52° above the horizontal with an initial speed of 36 m/sec. What is its range?
39. Grant Wishes is shot out of a cannon at 42° with an initial speed of 11.6 m/sec. How far away must he place the safety net?
40. A college freshman, frustrated with finals, releases his tensions by bombarding the adjacent dorm with water balloons. He fires one at 38° from the horizontal with a speed of 21.4 m/sec. How far up the building does it get? The dorm is 13.2 m away.
41. Since the great troglodyte war of previous pages, colony B has rebuilt and has developed its own technology. It can fire projectiles at different angles, though they all are released at v0 = 26 m/sec. How far from the top of the cliff will a bomb aimed 22° below the horizontal land?
42**. Where should troglodytes B aim to strike fear into those of colony A who live 88 meters below the surface? Remember, the projectile is fired at 26 m/sec.
You can find more general solutions to projectile problems by assigning variables rather than numerical values to the significant quantities.
43. A rock is kicked with velocity v off a cliff of height h. The
acceleration of gravity is g.
a) How long will it take to land?
b) How far from the base of the cliff will it land?
44. A disgruntled auto worker pushes a small foreign import off a 22 meter high cliff. If the vehicle lands 13 meters from the base, how fast was it pushed initially?
45. A creative chef cracks walnuts by catapulting them into a wall a distance s away.
If she releases them at a velocity v and angle q,
a) What are the initial horizontal and vertical velocities?
b) How long is the projectile in the air?
c) How high does it strike the wall?
46. A disgruntled fast food operator hurls an Egg McMuffin at 18 m/sec, 26° above
the horizontal, off a 12 meter high cliff.
a) What are its initial horizontal and vertical velocities?
b) How long will it be in the air?
c) How far from the cliff's base will it land?
47. A projectile is fired vertically up at 35 m/sec. How long will it take to reach the height of 40 m? Is it meaningful to have two solutions?
48. A projectile is fired up at 27 m/sec. How long will it be in flight if a bird catches it in mid air, while it is falling, 25 m from the ground?
49. A ball on a cliff is thrown vertically down at 12 ft/sec. If the cliff is 77 feet high, how long will it take to hit the ground?
50. A ball is thrown up at 19 ft/sec off the same 77 foot high cliff you learned to know and love in problem #49. How long will it take to hit the ground?
51. This time the ball is thrown off the edge of the cliff at 40 ft/sec, 28° above the horizontal. How long will it take to reach the ground? (The cliff is still 77 feet high.)
52. Now a ball is fired at 37 m/sec, 41° above the horizontal, up onto a cliff that is 26 m high. How far does it travel horizontally?
53. Finally, a ball is fired at 67° above the horizontal onto a cliff 38 m high, with an initial velocity of 32 m/sec. How far does it move horizontally?
54. A bridge is a height h above the ground.
a) How long will it take a rock to fall if released from rest off the bridge?
(gravity is g)
b) How long will it take if thrown down at velocity v0?
c) How long will it take if thrown up at v0?
55. A rock is thrown at velocity v0 at angle q on level ground.
a) What is its initial vertical velocity?
b) How long will it remain in the air?
c) How far will it get before hitting the ground?
56. Two cars head in the same direction. Car 1 moves with initial velocity v1 and accelerates at rate a. Car 2 moves with constant velocity v2, and has a head start of length L. When will the cars meet?
57. A rock is kicked horizontally off a tall cliff at velocity v0. What will its angle and speed be at time t?
58. A car accelerates at a constant rate a.
a) How long will it take to go a distance s?
b) How fast will it be going after traveling a distance s?
This equation, v2 = 2as, is the fourth and final kinematic equation. It, like the others, should be memorized. The four are:
s = vt s = 1/2 at2 v = at v2 = 2as
59*. A particle with initial velocity v0 accelerates at rate a. How fast will it be going after traveling a distance s?
60. Rosa Corn throws a banana vertically upward at 24 m/sec. How fast is it traveling at a height of 19 m above ground?
61. A champagne cork pops vertically upward at 29 ft/sec. To what maximum height will it rise? (Think What is its final velocity at the peak of its flight?)
62. At a race track a car is clocked at 15 ft/sec at one point, and at 47 ft/sec 40 feet farther down the track. What is its acceleration?
63. A bolt drops out of a bridge and falls 117 m to the river below. How fast is it traveling when it hits the water?
64. Arthur More watches the splash from the bolt in problem 63. He tries for a bigger splash by throwing a rock vertically downward at 27 m/sec. How fast will it strike the water?
Note that Arthur would get the same splash if he'd thrown the rock up at 27 m/sec.
65*. A bowling ball is flung at 106 ft/sec, 65° above the horizontal, striking a wall a distance L away. How high up the wall will it hit?
66*. Two walls are separated by a distance L. A projectile fired at velocity v0, downward at q as shown, lands a distance h below the top. Express q in terms of v0, L, h, and g.
Note that I have assigned positive values to all quantities in the vertical equation. Because gravity, initial velocity and final location are all in the same direction, they must be assigned the same sign -- all negative or all positive.
67**. Some happy college freshmen decide to have a water balloon fight in the hallway of their dorm. The ceiling is of height h, and the balloons are launched at velocity v0. At what angle will the balloons just graze the ceiling?