The Physics
Hypertextbook
Opus in profectus

# Impulse and Momentum

## Problems

### practice

1. Read the following excerpt of an interview with the American amateur naturalist Timothy Treadwell.

Now, the bears I live with average, the males, eight to twelve hundred pounds [360 to 540 kg]. They're the largest bears in the world…. They've been clocked at 41 [mph] and they've run a hundred meter dash in 5.85 seconds, which a human on steroids doesn't even approach.

1. Compute the speed of a grizzly bear using Mr. Treadwell's hundred meter statement.
2. Compute the momentum of a grizzly bear using the speed you calculated in part a. and the average mass stated by Mr. Treadwell.
3. How fast would a 250 lb man have to run to have the same momentum you calculated in part b? (Do not use a calculator to compute your answer.)
4. How fast would a 4000 lb car have to drive to have the same momentum you calculated in part b? (Do not use a calculator to compute your answer.)
2. The Indian Space Research Organisation (ISRO) launched the Mars Orbiter Mission from the Satish Dhawan Space Centre in Andhra Pradesh on 5 November 2013. The Mars Orbiter Spacecraft has been given the nickname मंगलयान (transliterated to Maṅgalayāna or Mnglyan), which is Sanskrit for "Mars craft". English speaking news agencies have been writing this as "Mangalyaan".

For the first month of its journey, the Mars Orbiter Spacecraft actually orbited the Earth. After a series of orbit raising maneuvers, the tiny main engine gave the spacecraft enough speed in the right direction to escape. For the next nine months the main engine was only used twice and only very briefly (less than a minute of total burn time) to correct the spacecraft's trajectory.

Nine months after leaving Earth orbit, the spacecraft arrived at Mars. To enter orbit around Mars, the spacecraft needed to slow down — a maneuver called orbit insertion. Since the main engine hadn't been used much in nine months, a quick little test was needed.

Press Release: September 22, 2014
Mars Orbiter Spacecraft's Main Liquid Engine Successfully Test Fired

The 440 newton Liquid Apogee Motor (LAM) of India's Mars Orbiter Spacecraft, last fired on December 01, 2013, was successfully fired for a duration of 3.968 seconds at 1430 hrs IST today (September 22, 2014). This operation of the spacecraft's main liquid engine was also used for the spacecraft's trajectory correction and changed its velocity by 2.18 metre/second. With this successful test firing, Mars Orbiter Insertion (MOI) operation of the spacecraft is scheduled to be performed on the morning of September 24, 2014 at 07:17:32 hrs IST by firing the LAM along with eight smaller liquid engines for a duration of about 24 minutes.

ISRO, 2014

1. Using the press release, determine the mass of the Mars Orbiter Spacecraft at the time of this test firing.
2. Why was the qualifying phrase "at the time of this test firing" added to the end of the previous question? Why does the mass of the spacecraft vary with time?
3. Read this passage about one of the difficulties of interstellar travel.

After all, the faster we go, the more difficult it is to avoid collisions with small objects and the more damage such a collision will wreak. Even if we are fortunate enough to miss all sizable objects, we can scarcely expect to miss the dust and individual atoms that are scattered throughout space. At two-tenths of the speed of light, dust and atoms might not do significant damage even in a voyage of 40 years, but the faster you go, the worse it is — space begins to become abrasive. When you begin to approach the speed of light, each hydrogen atom becomes a cosmic ray particle and will fry the crew. (A hydrogen atom or its nucleus striking the ship at nearly the speed of light is a cosmic ray particle, and there is no difference if the ship strikes the hydrogen atom or nucleus at nearly the speed of light. As Sancho Panza: "Whether the stone strikes the pitcher, or the pitcher strikes the stone, it is bad for the pitcher.") So 60,000 kilometers per second may be the practical speed limit for space travel.

Isaac Asimov, 1987

The density of the interstellar medium is about one hydrogen atom per cubic centimeter. Imagine a 1,000 tonne, 4 by 6 meter, classroom-sized interstellar spacecraft traveling at 60,000 km/s on its way to Proxima Centauri (the nearest solar system to our own) 4.243 light years away.

Kinematics:

1. How long would it take our hypothetical spacecraft to complete its hypothetical journey?

Impulse-Momentum:

1. Determine the momentum of our spacecraft.
2. What mass of interstellar medium is swept up during the journey?
3. What impulse does the interstellar medium deliver to the spacecraft?
4. How does this impulse compare to the momentum of the spacecraft?

Work-Energy:

1. Determine the kinetic energy of our spacecraft.
2. What is the effective drag force of the interstellar medium during the journey?
3. How much work does the interstellar medium do on the spacecraft?
4. How does this work compare to the kinetic energy of the spacecraft?
4. Write something.

### conceptual

1. In older passenger cars, body panels were attached to a single frame around the perimeter, making them very rigid. This is known as body-over-frame construction. In newer cars, different body parts have stress-bearing elements within them and these parts are then welded to each other. This is known as unitized body construction. Repairing "unibody" cars after collision is comparatively difficult as stress (and thus damage) are distributed throughout the different parts. Why then are cars now built this way
2. To escape from a horrible fire, two people are forced to jump from the third story of a burning building on to solid concrete. Which person is more likely to sustain serious injuries: the jumper who comes to an abrupt halt when he lands or the jumper who bounces after impact?
3. What is the basic idea behind crash safety features in cars like seatbelts, airbags, crumple zones, etc. What quantities in the impulse-momentum theorem (Ft = mv) change as a result of these features, how are they changed, and how does this result in increased safety during a crash?
4. The phrase "roll with the punches" has its origin in boxing. What does it mean to roll with a punch (or ride a punch)? How does rolling reduce the severity of a punch?
5. Is it possible for a motorcycle to have more momentum (that is, a momentum with a larger magnitude) than a train?

### numerical

1. When hit, the velocity of a 0.145 kg baseball changes from +20 m/s to −20 m/s. What is the magnitude of the impulse delivered by the bat to the ball?
2. A falling rubber ball of mass 0.025 kg strikes the ground traveling straight down at 4.0 m/s. Find the magnitude of the impulse that the ground gives to the ball if…
1. the ground is soft and the ball stops dead
2. the ground is hard and the ball bounces straight up at 2.0 m/s
3. A model rocket has mass of 1.5 kg. The engine exerts an effective upward thrust of 120 N for 3.2 seconds. (Assume a negligible amount of air resistance and no change in mass while the rocket is ascending.)
1. Draw a free body diagram showing all the forces acting on the model rocket.
Determine…
1. the weight of the rocket
2. the net force on the rocket while the engine was running
3. the net impulse on the rocket while the engine was running
4. the speed of the rocket when the engine stopped
5. the height of the rocket above the ground when the engine stopped
After the engine shuts down, the rocket is still moving upward.
1. Draw a free body diagram showing all the forces acting on the model rocket.
2. What is the acceleration of the rocket after the engine shut down?
3. What maximum height above the ground did the rocket reach?
4. A mechanic is pushing a disabled car. The graph below shows the force applied to the car by the mechanic vs. time.

1. What does the area under this curve represent?
2. Calculate its cumulative value at 2 s intervals. Compile your results in a table like the one below.
interval ending at 0 s 2 s 4 s 6 s 8 s 10 s
interval area
cumulative area
1. Sketch a graph of this quantity with respect to time.

### statistical

1. zarm-impulse.txt
The data in the accompanying tab delimited text file give the net force on a 500 kg projectile sitting atop a vertically mounted piston as a function of time. Use this data set and your favorite application for analyzing data to solve the following problems.
1. Use the given data to create a force-time graph.
2. Determine the impulse acting on the projectile as a function of time.
3. Compute the launch speed of the projectile.

Data adapted from Kampen, Kaczmarczik, and Rath; 2006.

### investigative

1. Which requires the bigger impulse: stopping the fastest pitched baseball completely or returning the fastest served tennis ball in the opposite direction with the same speed. To answer this question, complete a table like the one below. Provide the source of your info as well as its value. Use whatever units you find for mass and speed, but please report the impulse in newton seconds.
quantity baseball tennis ball
mass
(source)

speed
(source)

impulse
(show work)

2. Determine the momentum of one of Earth's tectonic plates (sometimes referred to as continental plates). A list of the most popular plates is shown on the right. Start with these three related questions.
1. Which plate have you chosen to work with?
2. What is the speed of this plate?
3. What is the area of this plate?
Now answer these three related questions.
1. Is the plate you've chosen continental or oceanic?
2. What is a typical density of this kind of crust?
3. What is a typical thickness of this type of crust?
Then finish the problem.
1. Compute the momentum of the tectonic plate you've chosen from the data you've found. State your answer to the nearest order of magnitude (the nearest power of ten). Don't forget the unit.
3. The National Association of Rocketry has a web page with links to data sheets for certified model rocket motors. Pick any one of these data sheets and find the following…
• the thrust-time data used to generate the graph on the data sheet
• the propellant mass
• the mass after firing (i.e., the mass of the empty rocket)
Calculate the following quantities as functions of time and make a time series graph of the…
1. impulse provided by the motor
2. fractional propellant mass loss (You need to determine the initial and final mass of the rocket. Assume that the loss of mass is directly proportional to the cumulative impulse. When the impulse is zero at the beginning, the mass loss is zero. When the impulse reaches its final value, the mass loss is 100%.)
3. mass of the rocket
4. speed of the rocket (Don't forget to include the force of gravity in your calculations.)
5. acceleration of the rocket
6. altitude of the rocket
Please note: because aerodynamic drag cannot easily be included into the calculations, you will wind up with final speeds that are too fast (but still within an order of magnitude of being "correct").
4. What's the momentum of… ? ☞ All data must be sourced. Bonus points for using primary sources. ☞ Show all work for any calculations including an equation and substitution with appropriate units.
event m
(kg)
v
(m/s)
p
(kg m/s)
1. swimmer, female, fastest 100 m freestyle
2. swimmer, male, fastest 100 m freestyle
3. sprinter, female, fastest 100 m dash
4. sprinter, male, fastest 100 m dash
5. softball, female, fastest pitch
6. Frisbee, fastest throw
7. baseball, male, fastest pitch
8. tennisball, fastest serve