The Physics
Hypertextbook
Opus in profectus

# Falling Bodies

## Problems

### practice

1. The following passages are excerpts from "The Long, Lonely Leap" by Captain Joseph W. Kittinger, Jr. USAF as they appeared in National Geographic magazine. It is the story of his record-setting, high altitude parachute jump from a helium balloon over New Mexico on 16 August 1960.

An hour and thirty-one minutes after launch, my pressure altimeter halts at 103,300 feet. At ground control the radar altimeters also have stopped on readings of 102,800 feet, the figure that we later agree upon as the more reliable. It is 7 o'clock in the morning, and I have reached float altitude.

At zero count I step into space. No wind whistles or billows my clothing. I have absolutely no sensation of the increasing speed with which I fall.

Though my stabilization chute opens at 96,000 feet, I accelerate for 6,000 feet more before hitting a peak of 614 miles an hour, nine-tenths the speed of sound at my altitude. An Air Force camera on the gondola took this photograph when the cotton clouds still lay 80,000 feet below. At 21,000 feet they rushed up so chillingly that I had to remind myself they were vapor and not solid.

Joseph Kittinger, 1960

Verify the speed claim of the author. (At this altitude g = 9.72 m/s2.)
2. A basketball dropped from rest 1.00 m above the floor rebounds to a height of 0.67 m. Assuming the ball is not moving horizontally, calculate its velocity…
1. just before it hit the floor on the way down and
2. just after it left the floor on the way up.
If the ball is in contact with the floor for 0.10 s determine its acceleration…
1. on the way down,
2. while it is contact with the floor, and
3. on the way up.
3. A diver jumps from a 3.0 m board with an initial upward velocity of 5.5 m/s. Determine…
1. the time the diver was in the air
2. the maximum height to which she ascended
3. her velocity on impact with the water
4. Sketch the following graphs of motion for an object thrown straight upward.
1. displacement-time
2. velocity-time
3. acceleration-time

### conceptual

1. A ball is thrown straight up over level ground. State the direction of the velocity and acceleration…
1. on the way up
2. at the highest point
3. on the way down

### numerical

1. Whole body reaction time is about three-tenths of a second. Would you have enough time to throw yourself clear of a brick falling from a point 3 m directly overhead if you saw it the moment it came loose?
2. A bullet leaves the muzzle of a 1.000 m long rifle with a velocity of 400.0 m/s when fired horizontally. Determine the muzzle velocity if the rifle is instead fired…
1. straight up
2. straight down
3. The terminal velocity of a skydiver is 55 m/s (120 mph) at typical jump altitudes.
1. Determine the minimum time and displacement needed to reach to reach this velocity for a skydiver starting from rest.
2. Why are these values minimums?
4. In the early 2000s, four skydivers proposed attempts to break Joseph Kittinger's 1960 world record parachute jump: Felix Baumgartner of Austria, Michel Fournier of France, Cheryl Stearns of the United States, and Rodd Millner of Australia. All planned to jump from an altitude of 40 km (130,000 feet) — 8 km (5 miles) higher than Captain Kittinger. With this additional distance, it is quite possible that one of them would have exceeded the speed of sound. Due to a series of technical and financial troubles, however, none of them have yet managed to get off the ground.
1. At what altitude might one of these skydivers break through the sound barrier? Assume that the acceleration due to gravity is 9.70 m/s2, the speed of sound is 300 m/s, and that air resistance is negligible.
2. Captain Kittinger believed that air resistance was negligible down to about 27.5 km (90,000 feet). Assuming a continued acceleration of 9.72 m/s2 after exceeding the speed of sound, determine a possible maximum speed during such a jump.
5. A baseball is thrown upward at 20 m/s. At what time is the ball…
1. 10 m above the point at which it was released?
2. 20 m above the point at which it was released?
3. 30 m above the point at which it was released?
6. A stone is thrown upward from a point 72 m above the ground and is airborne for 6 s.
1. Determine the initial velocity of the stone.
2. At some later time the stone is moving downward at 12 m/s.
1. When does this occur?
2. Where does this occur?
7. Two acrobats are about to perform a stunt; one on a trampoline and another 5.0 m above on a platform. At the instant that the acrobat on the platform steps off, the acrobat on the trampoline is moving upward at 7.5 m/s.
1. When do the two acrobats pass each other?
2. At what height above the trampoline are they?
3. What are their respective velocities?
8. This is a problem about not falling freely. Start by reading this passage…

Eight parachutists are to attempt a unique, low-altitude descent in September. They will leap from an altitude of just 75 m — lower than Big Ben but higher than Nelson's Column …. From exit to full inflation takes around 4 seconds …. That will leave around 30 m of descent with a full canopy in the 75 m drop. The entire descent is expected to take 10 seconds.

New Scientist, 1997

1. Determine the following quantities during the 4 s while the parachute is deploying…
1. the average vertical acceleration (magnitude and direction)
2. the final vertical speed (magnitude and direction)
2. Determine the following quantities during the 6 s while the parachute was fully deployed…
1. the average vertical acceleration (magnitude and direction)
2. the final vertical speed (magnitude and direction)
3. The "correct" solution to the last part of this problem gives a slightly nonsensical answer. Why does this happen?
9. Watch this video clipped from the 1978 movie version of Superman. Lois Lane falls from the heliport atop the Daily Planet building. Superman catches her and saves the day.

The fall

1. How long was Lois Lane in apparent free fall?
2. What speed would she have when Superman caught her?
3. How far would she have fallen in this time?
4. Describe the physical realism of this part of the scene.

The catch

1. How long did it take Superman to stop Lois Lane? (You may need to count frames. There are 25 frames per second in this video.)
2. Calculate Lois Lane's deceleration during the catch assuming she was traveling at the speed you calculated in part b.
3. Determine the distance needed to slow Lois Lane to a stop in the time given in part d.
4. Describe the likelihood of surviving a rescue attempt like this one.

A second opinion

1. Watch this video clipped from The Big Bang Theory, Season 1, Episode 2: The Big Bran Hypothesis (1 October 2007). Was Sheldon right about Lois Lane's chances of surviving a fall like the one in the movie?

10. Skateboarder Jake Browning plummeted to the floor so hard in the 2007 X Games his shoes flew off. Watch the video below to see what happened (and to get some insight from the newscasters.)
1. How many seconds did it take Mr. Browning to fall from his highest position until he just touched the floor?
2. What speed did he have on impact?
3. What total height did he fall?
4. How long did it take him to stop after he touched the floor?
5. What was his deceleration?
6. Ouch?
11. Watch the Blob Jump Official Guinness World Record video below. Disregard the horizontal motion (i.e., assume the blob jumper went straight up and then straight back down) and answer the following questions.
1. When did the blob jumper reach the highest point of the jump? How long did it take him to hit the lake surface after reaching the highest point of the jump? Use these numbers to answer the remaining parts of this problem.
2. How did his velocity on take off compare to his velocity on landing? Determine both velocities (their magnitudes and directions).
3. How high did he rise? Confirm or refute the claim that the jumper hit 17 m.
4. Fun?

### investigative

1. You can determine a person's reaction time using a centimeter ruler. Find several volunteers and have them hold their open hand out so that you can drop a ruler between their thumb and fingers. Suspend the ruler vertically with the zero mark of the ruler at the same level as the top of your volunteer's open hand. Tell your volunteers to grab the ruler the instant you drop it. Release it without warning and record the position on the ruler where they grabbed it. (Take the reading from the top of their fingers.) Repeat this test a few times for each volunteer to obtain an average value. Record your results along with your volunteer's ages (or another demographic variable that you think may be relevant). Determine the relation, if any, between age and reaction time in your sample. (A list of demographic variables to test might include such things as gender, place of birth, hours spent playing video games, hours spent watching TV, number of siblings, whatever.)
2. How high should a domed baseball stadium be built if it is to accommodate even the highest pop fly? There are two ways to solve this problem…
1. using the time a pop fly is in the air or
2. using the speed of a batted ball.
Use whichever method you find easiest.