Second law as rate of momentum change

Intermediate

Force is the time rate of change of momentum.

Key relationF = dp/dt

Newton's original statement is F = dp/dt. It reduces to ma for constant mass but also covers changing mass (rockets).

Worked example

A force of 10 N acting for 3 s changes the momentum by 30 kg·m/s.

Common mistake

F = ma only when m is constant.

In more detail

Newton's second law can be written as the net force equals the rate of change of momentum. This form also applies to systems where mass changes.

Engineers use this form for rockets, where mass is continuously expelled.
Jet engines and water jets are analysed using the rate of change of momentum.
Car crash tests measure how the change in momentum over time shapes the force on the occupants.

Why is this form more general?

It works even when the mass of the system changes, while F = ma does not.

Does it give the same answer for constant mass?

Yes. For constant mass it reduces to F = ma.

Why is the time rate of momentum change useful?

It gives the force directly for changing momentum, including cases where the mass also changes.