Impulse and the Impulse-Momentum Theorem

Newton's second law is really F = dp/dt, force equals the rate of change of momentum. Rearrange and integrate over time and you get the impulse-momentum theorem: J = integral of F dt = delta p = m·delta v. The left side, impulse, is force accumulated over time. The right side is the actual physical result, the change in momentum. These are always equal, no matter how the force varies during the interaction.

This is why stretching contact time matters. A given change in momentum is fixed by the situation, how fast something was moving and how fast it ends up moving. If that change in momentum has to happen over a longer time, the average force needed is smaller, because average force equals delta p divided by delta t. A mattress extends your stopping time compared to concrete, so the same change in your momentum needs far less force to produce it. That smaller force is what your body actually feels.

This connects directly to collisions, the next idea in the unit. Impulse-momentum theorem works on any single object regardless of what it hits, while momentum conservation looks at a pair of objects during a collision and uses Newton's third law to say their impulses on each other are equal and opposite. Impulse is also the area under a force-versus-time graph, which is how you handle forces that are not constant, like the spiky force curve during a real bat-ball collision.