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# Momentum: Of Special Interest to Physics C Students

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By McGraw-Hill Professional
Updated on Feb 10, 2011

Practice problems for these concepts can be found at: Momentum Practice Problems for AP Physics

### Calculus Version of the Impulse–Momentum Theorem

Conceptually, you should think of impulse as change in momentum, also equal to a force multiplied by the time during which that force acts. This is sufficient when the force in question is constant, or when you can easily define an average force during a time interval.

But what about when a force is changing with time? The relationship between force and momentum in the language of calculus is

A common AP question, then, gives momentum of an object as a function of time, and asks you to take the derivative to find the force on the object.

It's also useful to understand this calculus graphically. Given a graph of momentum vs. time, the slope of the tangent to the graph gives the force at that point in time. Given a graph of force vs. time, the area under that graph is impulse, or change in momentum during that time interval.

### Motion of the Center of Mass

The center of mass of a system of objects obeys Newton's second law. Two common examples might illustrate the point:

1. Imagine that an astronaut on a spacewalk throws a rope around a small asteroid, and then pulls the asteroid toward him. Where will the asteroid and the astronaut collide?
2. Answer: at the center of mass. Since no forces acted except due to the astronaut and asteroid, the center of mass must have no acceleration. The center of mass started at rest, and stays at rest, all the way until the objects collide.

3. A toy rocket is in projectile motion, so that it is on track to land 30 m from its launch point. While in the air, the rocket explodes into two identical pieces, one of which lands 35 m from the launch point. Where does the first piece land?
4. Answer: 25 m from the launch point. Since the only external force acting on the rocket is gravity, the center of mass must stay in projectile motion, and must land 30 m from the launch point. The two pieces are of equal mass, so if one is 5 m beyond the center of mass's landing point, the other piece must be 5 m short of that point.

### Finding the Center of Mass

Usually the location of the center of mass (cm) is pretty obvious … the formal equation for the cm of several objects is

Mxcm = m1x1 + m2x2 + …

Multiply the mass of each object by its position, and divide by the total mass M, and voila, you have the position of the center of mass. What this tells you is that the cm of several equal-mass objects is right in between them; if one mass is heavier than the others, the cm is closer to the heavy mass.

Very rarely, you might have to find the center of mass of a continuous body (like a baseball bat) using calculus. The formula is

Do not use this equation unless (a) you have plenty of extra time to spend, and (b) you know exactly what you're doing. In the highly unlikely event it's necessary to use this equation to find a center of mass, you will usually be better off just guessing at the answer and moving on to the rest of the problem. (If you want to find out how to do such a problem thoroughly, consult your textbook. This is not something worth reviewing if you don't know how to do it already.)

Practice problems for these concepts can be found at: Momentum Practice Problems for AP Physics

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