# Mechanics

calculus-based physics for scientists and engineers

## Conservation of Linear Momentum

### Conservation of Linear Momentum

What we often just call momentum, is more precisely called linear momentum. The best way to describe the conservation of linear momentum is to say: if the net external force along any linear direction is zero for a system, the component of the total linear momentum for the system along that direction is conserved. That may be a bit confusing. If the total net external force is zero, the idea simplifies a bit. If the total net force is zero on a system, the total linear momentum of the system is conserved. The first key point is that at issue is the system. If you have multiple objects, you are interested in the net external force on the entire system. The second key point is external. You don't care about internal forces. You only need to consider forces whose agent is not part of the system. The third key point is the total linear momentum is conserved. That means you have to add up the momenta of all the components of the system. This is a vector sum, since momentum is a vector. However, let us return to the beginning. This criteria holds for any linear dimension of the system. You don't need the total net external force to be zero to use conservation of momentum. If the net external force is zero along any linear dimension, then the component of the total linear momentum is conserved along that dimension.

### Conservation of Linear Momentum: Example

Let's look at an example using conservation of linear momentum.

### The Impulse Approximation

The requirement that no net external force acts on the system greatly restricts the problems in which conservation of momentum can be used. However, there are many times when momentum conservation can be used to solve problems even though there is a net external force. This happens when the external force is small compared to the internal forces and the time interval of interest is very short. This is called the impulse approximation.

### Solution

1. A 10-kg block sits on the ground motionless. A 0.01-kg bullet moving at 100 m/s hits the block and goes all the way through it very quickly, exiting at 60 m/s. In doing so, it removes 0.2 kg of the block and propels this fragment forward at 20 m/s. What is the resulting speed and direction of the block? Assume everything is taking place in one dimension.

### Solution

2. A 1-kg block sits on the ground motionless. A 0.01-kg steel bullet moving at 100 m/s hits the block at sticks into it. Repeating the experiment, a 0.01-kg rubber bullet moving at 100 m/s hits the block and bounces back at about the same speed. In which situation does the block recoil more?