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Mechanics

The Relation Between Force and Momentum

Consider a body of mass m, initially moving with a velocity of magnitude u. A force F acts on the body and causes it to accelerate to a final velocity of magnitude v.

We can write Newton’s second law in the form

and a simple rearrangement shows the relation between force and momentum

Now, mv is the final momentum of the body and mu is the initial momentum of the body.

Therefore, we have

Force = rate of change of momentum

So, alternative units for momentum are Ns.

Rearranging this equation gives

change in momentum, DELTA01p = FDELTA01t

The quantity FDELTA01t is called the impulse of the force F.

If the force acting on the body is not constant we can write

DELTA01p = average force × DELTA01t

Suppose the force acting on a body varies as shown by the graph below.

During the first three seconds the change in momentum was

DELTA01p1 = 4 × 3 = 12 Ns

During the next four seconds the change in momentum was

DELTA01p2 = 8 × 6 = 48 Ns

So the total change in momentum was 60 Ns.

N.B.

Notice that the change in momentum can be found by calculating the area under the graph of force against time.

This can be useful if the graph does not consist of straight lines.

In situations where a continuous flow of matter occurs, for example, water flowing out of a pipe, gas flowing out of a rocket or jet engine etc, it is often useful to consider a slight rearrangement of the equation relating force to momentum.

 

Therefore we can write

 

So, if we know the rate of flow of mass, and the change in velocity of each particle we can calculate the force needed to cause the change in momentum.

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