Force is the rate of change of momentum

5 minAS Level3.1.43.1.5

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can define force as the rate of change of momentum, use $F = \Delta p / \Delta t$ , state Newton's second law in full, and show where $F = ma$ comes from.

Why you bend your knees

Jump off a wall and land stiff-legged: ouch. Land and bend your knees: no problem. Either way your momentum has to drop to zero.

The difference is how quickly it drops. And "how quickly momentum changes" is, it turns out, exactly what a force is.

Definition

Force

Force is the rate of change of momentum:

$F = \frac{\Delta p}{\Delta t}$

If the force varies during $\Delta t$ , this gives the average resultant force.

Definition

Newton's second law

The rate of change of momentum of an object is proportional to the resultant force acting on it, and the change in momentum takes place in the direction of that force.

The newton is defined so that the constant of proportionality is 1, which gives $F = \dfrac{\Delta p}{\Delta t}$ .

Where F = ma comes from

If the mass stays constant, only the velocity changes:

$F = \frac{\Delta (mv)}{\Delta t} = m\,\frac{\Delta v}{\Delta t} = ma$

So $F = ma$ is the second law for constant mass. The momentum version is more general: it still works when mass flows in or out, like a rocket burning fuel or a hose spraying water.

Quick check

DefinitionQuestion 1 · Recall

Fill in the blanksQuestion 2 · Recall

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