Momentum and Newton's Second Law

Newton did not write F = ma. He wrote that force equals the rate of change of momentum, F = dp/dt, where p = mv. Momentum is the quantity that tracks both how much mass is moving and how fast. Writing the law this way makes it work even when mass itself is changing, not just velocity. That generality is the whole point: force is whatever changes motion, and momentum is the full measure of motion.

When mass is constant, dp/dt = d(mv)/dt = m(dv/dt) = ma, since the derivative only has to act on v. The m slides out front untouched. That is why F = ma feels like the whole story: in nearly every intro problem, mass is fixed, so the two forms agree perfectly. But F = dp/dt is the deeper statement, and it is the one that still works when mass is not fixed, like a rocket burning fuel or a raindrop growing as it falls through a cloud.

This sets up the rest of the unit. Impulse, J = integral of F dt = delta p, is just this same law integrated over time, and it is how you handle forces that vary or act briefly, like a bat hitting a ball. Momentum conservation in collisions follows from the same root idea: with no external force, dp/dt = 0 for the whole system, so total momentum cannot change even while it shuffles between colliding objects.