Momentum and impulse

Motion that is hard to stop

Momentum is a measure of how much motion an object has. A slow-moving bicycle has some momentum. A fast-moving truck has much more. Momentum depends on both mass and velocity:

$p = mv$

Here $p$ is momentum, $m$ is mass, and $v$ is velocity. Because velocity includes direction, momentum also has direction. A car moving east has eastward momentum; the same car moving west has westward momentum.

Momentum helps explain why some moving objects are harder to stop than others. A heavy object and a fast object both tend to have large momentum. To stop them, their momentum must be changed.

Force changes momentum

Newton's second law can be written in terms of momentum. A net force changes momentum over time. The larger the force, or the longer it acts, the greater the change in momentum.

Impulse is the product of force and time:

$J = FDelta t$

Impulse equals change in momentum:

$J = Delta p$

This means the same change in momentum can happen with a large force over a short time or a smaller force over a longer time.

Safety and stopping time

Impulse is central to safety design. When a car stops suddenly in a crash, the passengers' momentum must change to zero. If that change happens in a very short time, the force is large and dangerous. Seat belts, airbags, helmets, padded mats, and crumple zones increase the stopping time and distance, reducing the force on the body.

For example, jumping onto a stiff floor hurts more than landing on a padded mat because the mat extends the stopping time. Your momentum changes by the same amount in both cases if you come to rest, but the force is smaller when the stopping time is longer.

Follow-through in sports

Impulse also appears in sports. A baseball bat, tennis racket, or golf club changes the ball's momentum. Following through can increase the contact time and help transfer momentum effectively. Catching a ball by moving your hands backward also increases stopping time, reducing the force on your hands.

These examples show that impulse is not only about collisions. It is about any situation where force acts for a time to change motion.

Momentum conservation

Momentum is especially powerful because total momentum is conserved in a closed system with no net external force. This means the total momentum before an interaction equals the total momentum after it.

For two interacting objects:

$p_{total,before} = p_{total,after}$

This idea explains recoil, collisions, explosions, and motion in space. When a rocket pushes exhaust gases backward, the gases gain momentum one way and the rocket gains momentum the other way. The total momentum of the rocket-gas system is conserved.

Momentum versus energy

Momentum and energy are related but different. Kinetic energy depends on $v^2$, while momentum depends on $v$. Energy is a scalar, meaning it has size but not direction. Momentum is a vector, meaning direction matters.

In collisions, momentum is conserved when external forces are negligible. Kinetic energy may or may not be conserved, depending on the type of collision. This distinction becomes important when studying crashes and rebounds.

The big idea

Momentum measures motion with direction. Impulse measures the effect of force over time. Increasing the time over which momentum changes can reduce force, which is why safety devices work. Conservation of momentum gives a powerful way to analyze interactions even when the forces during the interaction are complicated.