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9.1 Linear momentum

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By the end of this section, you will be able to:
  • Explain what momentum is, physically
  • Calculate the momentum of a moving object

Our study of kinetic energy showed that a complete understanding of an object’s motion must include both its mass and its velocity ( K = ( 1 / 2 ) m v 2 ). However, as powerful as this concept is, it does not include any information about the direction of the moving object’s velocity vector. We’ll now define a physical quantity that includes direction.

Like kinetic energy, this quantity includes both mass and velocity; like kinetic energy, it is a way of characterizing the “quantity of motion” of an object. It is given the name momentum    (from the Latin word movimentum , meaning “movement”), and it is represented by the symbol p .

Momentum

The momentum p of an object is the product of its mass and its velocity:

p → = m v → .
Photo of a soccer player kicking a ball. Two arrows have been added to the photo at the ball’s position. Both arrows point forward, in the direction the player is kicking. One arrow is labeled velocity, the other arrow is labeled momentum.
The velocity and momentum vectors for the ball are in the same direction. The mass of the ball is about 0.5 kg, so the momentum vector is about half the length of the velocity vector because momentum is velocity time mass. (credit: modification of work by Ben Sutherland)

As shown in [link] , momentum is a vector quantity (since velocity is). This is one of the things that makes momentum useful and not a duplication of kinetic energy. It is perhaps most useful when determining whether an object’s motion is difficult to change ( [link] ) or easy to change ( [link] ).

A photo of a supertanker in the water is shown. There are two much smaller vessels with sails in the distance.
This supertanker transports a huge mass of oil; as a consequence, it takes a long time for a force to change its (comparatively small) velocity. (credit: modification of work by “the_tahoe_guy”/Flickr)
A drawing of a stoppered flask, labeled “container”, with gas molecules (represented as green dots) moving randomly inside the flask.
Gas molecules can have very large velocities, but these velocities change nearly instantaneously when they collide with the container walls or with each other. This is primarily because their masses are so tiny.

Unlike kinetic energy, momentum depends equally on an object’s mass and velocity. For example, as you will learn when you study thermodynamics, the average speed of an air molecule at room temperature is approximately 500 m/s, with an average molecular mass of 6 × 10 −25 kg ; its momentum is thus

p molecule = ( 6 × 10 −25 kg ) ( 500 m s ) = 3 × 10 −22 kg · m s .

For comparison, a typical automobile might have a speed of only 15 m/s, but a mass of 1400 kg, giving it a momentum of

p car = ( 1400 kg ) ( 15 m s ) = 21,000 kg · m s .

These momenta are different by 27 orders of magnitude, or a factor of a billion billion billion!

Summary

  • The motion of an object depends on its mass as well as its velocity. Momentum is a concept that describes this. It is a useful and powerful concept, both computationally and theoretically. The SI unit for momentum is kg · m/s.

Conceptual questions

An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy?

Since K = p 2 / 2 m , then if the momentum is fixed, the object with smaller mass has more kinetic energy.

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An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?

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Problems

An elephant and a hunter are having a confrontation.

A drawing of an elephant, on the left, and hunter, on the right. An x y coordinate system is shown, with positive x to the right and positive y up. The elephant is labeled with m E = 2000.0 k g, and vector v E = 7.50 meters per second times I hat. An arrow above the v E vector points to the right. The hunter is labeled with m hunter = 90.0 k g, and vector v hunter = 7.40 meters per second times I hat. An arrow above the v hunter vector points to the right. Between the hunter and elephant is a dart with a long arrow pointing to the left drawn near it and labeled vector v dart = 600 meters per second times minus I hat, and m dart = 0.0400 k g.
  1. Calculate the momentum of the 2000.0-kg elephant charging the hunter at a speed of 7.50 m/s.
  2. Calculate the ratio of the elephant’s momentum to the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s.
  3. What is the momentum of the 90.0-kg hunter running at 7.40 m/s after missing the elephant?
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A skater of mass 40 kg is carrying a box of mass 5 kg. The skater has a speed of 5 m/s with respect to the floor and is gliding without any friction on a smooth surface.

  1. Find the momentum of the box with respect to the floor.
  2. Find the momentum of the box with respect to the floor after she puts the box down on the frictionless skating surface.

a. magnitude: 25 kg · m/s; b. same as a.

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A car of mass 2000 kg is moving with a constant velocity of 10 m/s due east. What is the momentum of the car?

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The mass of Earth is 5.97 × 10 24 kg and its orbital radius is an average of 1.50 × 10 11 m . Calculate the magnitude of its average linear momentum.

An illustration of the earth orbiting the sun. The mass of the earth is given as 5.97 times 10 to the 24 kilograms and the radius of the orbit is labeled R earth = 1.5 times 10 to the 11 meters.

1.78 × 10 29 kg · m/s

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If a rainstorm drops 1 cm of rain over an area of 10 km 2 in the period of 1 hour, what is the momentum of the rain that falls in one second? Assume the terminal velocity of a raindrop is 10 m/s.

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What is the average momentum of an avalanche that moves a 40-cm-thick layer of snow over an area of 100 m by 500 m over a distance of 1 km down a hill in 5.5 s? Assume a density of 350 kg/m 3 for the snow.

1.3 × 10 9 kg · m/s

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What is the average momentum of a 70.0-kg sprinter who runs the 100-m dash in 9.65 s?

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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