The Physics
Hypertextbook
Opus in profectus

# Free Fall

## Problems

### practice

1. The following passages are excerpts from "The Long, Lonely Leap" by Captain Joseph Kittinger 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
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
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 up.
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 of the ball after it has left the thrower's hand and is…
1. on the way up
2. at it 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 (200 km/h, 120 mph) at typical jump altitudes.
1. Determine the minimum time needed to reach to reach this velocity for a skydiver starting from rest.
2. Determine the minimum displacement needed to reach to reach this velocity for a skydiver starting from rest.
3. 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. At what time does this occur?
2. At what height 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. ZARM! Despite my addition of an exclamation point, it's not an onomatopoeic sound effect from a comic book. Das Zentrum für Angewandte Raumfahrttechnologie und Mikrogravitation (or in English, the Center for Applied Space Technology and Microgravity) is a research institute at the University of Bremen in Germany that studies space technology and other sciences as they relate to gravity. The most prominent structure at this facility is the nearly 150 meter tall drop tower. Gravity is a fundamental force that acts on all objects everywhere in the universe. While it is impossible to eliminate gravity, its effects on systems can be reduced during free fall — a state often described as weightlessness, but that scientists prefer to call microgravity. In any case, when gravity gets turned off, things get interesting. At ZARM, scientists can choose between one of two different flight campaigns in the drop tower. The first is "drop mode" — a straight free fall.

During a simple drop experiment, the capsule is pulled up to a height of 120 meters to the top of the drop tube and then released. After 4.74 seconds the experiment has landed safely in the deceleration unit filled with polystyrene pellets. Before the experiment, 18 high-performance pumps make sure that the drop tube is almost free of air containing only one ten thousandth of the normal air pressure. Due to the vacuum, the air drag is so low that the Bremen Drop Tower can provide one of the best quality of microgravity — in some aspects even better than on the International Space Station (ISS). Therefore, ZARM's drop tower facility is a very economic and easily accessible alternative to doing research in space.

ZARM, 2020

Answer the following questions about a drop mode experiment using the information provided in the extended quote above.
1. Over what distance does the capsule fall freely? (This is not the same as the height of the drop tube or the height of the tower.)
2. What is the impact speed of a capsule?
3. What is the length of the deceleration unit?
4. What is the average acceleration (magnitude and direction) of a capsule in the deceleration unit?
The second type of flight campaign is "catapult mode", where the capsule is thrown vertically up from the bottom and then caught when it returns.

The pneumatically driven system takes 0.25 seconds to accelerate the experiment capsule to a speed of 168 kilometers per hour. The exact force of acceleration is being calculated for each individual experiment in order to throw the drop capsule as close as possible to the top of the drop tube and thus maximize the duration of the flight. After a couple of seconds the deceleration unit has already been moved into place again in order to catch the capsule on its way down.

ZARM, 2020

Answer the following questions about a catapult mode experiment using the information provided in the extended quote above.
1. What is the launch speed of the capsule in meters per second?
2. What is the average acceleration (magnitude and direction) of the capsule while it is being pushed up in the catapult?
3. Over what distance does the catapult push a capsule?
4. For how long is a capsule in free fall after a launch? Or in other words, how long does it take a capsule to return to the height from which it was launched?
5. How high does the capsule rise after being lauched from the catapult? (This is not the same as the height of the drop tube or the height of the tower or the answer to any previous part of this question.)
9. Lois Lane is in a helicopter departing from the roof of the Daily Planet Building where she works in the great American city of Metropolis. There is a problem with the flight and the helicopter is caught on a cable before it can leave the rooftop heliport. The helicopter skids to the edge of the building and the door opens. Lois falls until she reaches a top speed of 60 m/s. Just in the nick of time, Superman swoops up from the street below and catches her.
1. How far did Lois Lane fall before Superman caught her?
2. For how long was Lois Lane falling?
After Superman catches Lois Lane, they slow down safely together with a deceleration of −80 m/s2. Superman and Lois Lane come to a complete stop the instant they touch the sidewalk. Due to the rapid deceleration Lois Lane temporarily loses consciousness, but she is not hurt. Superman's hair is slightly disheveled and he does not have a comb.
1. How long did it take Superman and Lois Lane to stop?
2. How far did they travel while stopping?
3. How tall is the Daily Planet Building?
10. The diagram to the right shows a US one dollar bill held vertically above a person's hand with their finger and thumb arranged like they are ready to pinch something. Fingertip reaction time is about 200 ms for an average person. Show that it is unlikely that someone would be able to catch a US dollar bill in a situation like this if it was released without warning.

### 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.
3. 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 clip 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?
4. Skateboarder Jake Brown 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. Brown to fall from his highest position until he just touched the floor?
2. What speed did he have on impact?
3. What height did he fall?
4. How long did it take him to stop after he touched the floor?
5. What was his average deceleration?
6. Ouch?
5. 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?
6. The video below shows a rock being dropped from a bridge to a creek below.

1. For how long was the rock falling? (The playback speed is 14.81 frames per second.)
2. How far did the rock fall?
3. What speed did the rock have when it struck the water?
4. How long did it take the rock to fall half the distance you calculated in part b?
5. How long did it take the rock to reach half the speed you calculated in part c?
After striking the water, the rock came to rest in 30 cm.
1. Determine the deceleration of the rock after it struck the water.
2. Determine the time it took the rock to stop.
7. The Chicago Fire Department released this amazing video of a black cat leaping from the fifth story window of a building that was on fire. It landed on grass, bounced once, and walked away — apparently unhurt. Use video analysis to determine…
1. the height the cat fell
2. the speed it had when it hit the grass
(Since downloading media from social media websites is not always easy, a copy of the video has been placed on this website for your convenience.)