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 



mv is the final momentum of the
body and mu is the initial momentum, therefore the relation between
force and momentum is: 

force = rate of
change of momentum 

usually written as 



rearranging the equation gives 



From the definition of momentum (mass×velocity)
we see that the units are kgms^{1}
but the above equation shows that alternative and, of course,
equivalent, units for momentum are Newtons seconds,
Ns 



The quantity ∆p (change in momentum) is called the
impulse of the force causing the change. 



If the force acting on the body is not constant, we can still use the above
equation to find the change in momentum, assuming that we can find the
average value of the force. 



Suppose that the force acting
on a body varied as shown in the graph below. 




During the first three seconds the average
force was 4N so the change in momentum was 

4 × 3
= 12Ns 



During the next
four seconds the change in momentum was 

8 × 6 = 48Ns 





So the total change in momentum was 60Ns
(or 60kgms^{1}
if you prefer) 



Notice that the calculation of the change in momentum is equivalent to
finding 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 force (or pressure) causes a continuous flow of matter
to occur, 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: 



So, if we wish to find, for example, the force exerted by a jet engine (or
similar) we just need to know the rate of flow of matter in the jet
(number of kg of mass ejected per second) and the change in velocity
of the matter being ejected. 
