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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 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 re-arranging 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.
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