How to Calculate Force from Impulse and Momentum

In physics, you can use the impulse-momentum theorem to calculate force based on impulse and momentum. For example, you can relate the impulse with which you hit an object to its consequent change in momentum. According to the theorem:

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How about using the equation the next time you hit a pool ball? You line up the shot that the game depends on. You figure that the end of your cue will be in contact with the ball for 5 milliseconds (a millisecond is a thousandth of a second).

You measure the ball at 200 grams (or 0.200 kilograms). After testing the side cushion with calipers, spectroscope, and tweezers, you figure that you need to give the ball a speed of 2.0 meters per second. What average force do you have to apply? To find the average force, first find the impulse you have to supply. You can relate that impulse to the change in the ball’s momentum this way:

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Because the pool ball doesn’t change direction, you can use this equation for the component of the pool ball’s momentum in the direction in which you strike it. Because you’re using a component of the vector, you remove the bold from p.

So what’s the change in the ball’s momentum? The speed you need, 2.0 meters per second, is the magnitude of the pool ball’s final velocity. Assuming the pool ball starts at rest, the change in the ball’s momentum will be the following:

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where v is the component of the ball’s velocity in the direction in which you strike it. Plugging in the numbers gives you the change in momentum:

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You need a change in momentum of 0.40 kilogram-meters per second, which is also the impulse you need. Because

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this equation becomes the following for the component of the force in the direction of motion:

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Therefore, the force you need to apply works out to be

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In this equation, the time your cue ball is in contact with the ball is 5 milliseconds, or

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Plug in the time to find the force:

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You have to apply about 80 newtons (or about 18 pounds) of force, which sounds like a huge amount. However, you apply it over such a short time, 


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that it seems like much less.

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