# Momentum

## Linear Momentum, Force, and Impulse


### Section Key Terms


### Momentum, Impulse, and the Impulse-Momentum Theorem

Linear momentum is the product of a system’s mass and its velocity. In equation form, linear momentum p is
You can see from the equation that momentum is directly proportional to the object’s mass (m) and velocity (v). Therefore, the greater an object’s mass or the greater its velocity, the greater its momentum. A large, fast-moving object has greater momentum than a smaller, slower object. 
Momentum is a vector and has the same direction as velocity v. Since mass is a scalar, when velocity is in a negative direction (i.e., opposite the direction of motion), the momentum will also be in a negative direction; and when velocity is in a positive direction, momentum will likewise be in a positive direction. The SI unit for momentum is kg m/s.
Momentum is so important for understanding motion that it was called the quantity of motion by physicists such as Newton. Force influences momentum, and we can rearrange Newton’s second law of motion to show the relationship between force and momentum.
Recall our study of Newton’s second law of motion (Fnet = ma). Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. The change in momentum is the difference between the final and initial values of momentum.
In equation form, this law is
where Fnet is the net external force,  
  is the change in momentum, and  
  is the change in time.
We can solve for  
  by rearranging the equation  
to be

 

is known as impulse and this equation is known as the impulse-momentum theorem. From the equation, we see that the impulse equals the average net external force multiplied by the time this force acts. It is equal to the change in momentum. The effect of a force on an object depends on how long it acts, as well as the strength of the force. Impulse is a useful concept because it quantifies the effect of a force. A very large force acting for a short time can have a great effect on the momentum of an object, such as the force of a racket hitting a tennis ball. A small force could cause the same change in momentum, but it would have to act for a much longer time. 

### Newton’s Second Law in Terms of Momentum

When Newton’s second law is expressed in terms of momentum, it can be used for solving problems where mass varies, since  
 . In the more traditional form of the law that you are used to working with, mass is assumed to be constant. In fact, this traditional form is a special case of the law, where mass is constant. 
  is actually derived from the equation:
For the sake of understanding the relationship between Newton’s second law in its two forms, let’s recreate the derivation of  
  from
by substituting the definitions of acceleration and momentum.
The change in momentum  
  is given by
If the mass of the system is constant, then
By substituting  
  for  
, Newton’s second law of motion becomes 
for a constant mass.
Because
we can substitute to get the familiar equation
when the mass of the system is constant.

### Solving Problems Using the Impulse-Momentum Theorem


### Practice Problems


### Check Your Understanding


### Section Summary

1. Linear momentum, often referenced as momentum for short, is defined as the product of a system’s mass multiplied by its velocity, p = mv.
2. The SI unit for momentum is kg m/s.
3. Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes,  
 .
4. Impulse is the average net external force multiplied by the time this force acts, and impulse equals the change in momentum,  
 .
5. Forces are usually not constant over a period of time, so we use the average of the force over the time it acts.

### Key Equations


### Concept Items


### Critical Thinking


### Problems


### Test Prep Multiple Choice


### Test Prep Short Answer


### Test Prep Extended Response
