Impulse is equal to the change in momentum of an object. This relationship is expressed by the impulse-momentum theorem:
Impulse is equal to the change in momentum of an object. This relationship is expressed by the impulse-momentum theorem:
Here, is impulse, is initial momentum, and is final momentum.
Momentum is the product of mass and velocity:
Newton's second law states that the resultant force equals the rate of change of momentum:
Rearranging gives:
For a constant force, impulse is . Therefore, applying a force over a time interval changes an object's momentum by the same amount. Both impulse and momentum are vector quantities, so direction must be included.
For example, a football initially at rest is kicked until it moves at :
The impulse delivered to the ball is therefore in the direction of motion. The units are equivalent because .
| Situation | Effect on impulse and momentum |
|---|---|
| Greater force for the same contact time | Greater impulse and greater change in momentum |
| Longer contact time for the same force | Greater impulse and greater change in momentum |
| Same impulse delivered over a longer time | Same change in momentum but smaller average force |
When force varies, average force can be used: . On a force-time graph, each narrow strip represents , so the total signed area gives impulse. Area above the time axis is positive and area below it is negative according to the chosen direction.
This is especially relevant to sports safety. During landing, bending the knees increases the time over which the athlete's momentum falls to zero. The change in momentum is unchanged, but decreases, reducing the average resultant force on the body.
In IB SEHS B.2.1, define momentum, state , show substitutions with units, and indicate direction. A common misconception is that increasing contact time always increases impulse. It does so only if force is unchanged; for a fixed momentum change, increasing contact time reduces average force.