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External Forces and Impulse

So what happens if there is an unbalanced external force? There must be a change of momentum:


If the external forces on an object add up to Fnet, then we can write


The change in momentum Δp is called the impulse. So the impulse and the net force on a system are related by


or in one dimension


Whenever you see a problem involving force and time, you should think about momentum and write this equation.
Why is bouncing on a trampoline less painful than bouncing off a cement sidewalk?
  1. The force is less because the mass of the sidewalk is greater.
  2. The force is less because the area of the sidewalk is greater.
  3. The force is less because the time of impact is less.
  4. The force is less because the time of impact is greater.

Choice A reminds us of the equation Fnet = ma, so let's see if that makes any sense. If the mass of the sidewalk is large, then the sidewalk's acceleration will be small, for a given force. Or else the force the sidewalk must experience for a given acceleration is large. It's hard to see how this makes a difference.
Choice B reminds us of the definition of pressure, so let's write the equation P = F/A. There is no way to reconcile choice B with this equation, however, since greater area implies a greater force, for a given pressure.
Choices C and D remind us of equation (5) above, so we write Δp = Fnet Δt. The force would be less for greater Δt, if the impulse is constant. Can we make sense of this choice? The impulse, or change in momentum, is the same in either case, since the body goes from moving downward to moving upward, and Δp = mΔv. Also, because of the elasticity of the trampoline, a body is in contact with it for a longer time. So choice D makes sense.
This question is reminiscent of many problems on the MCAT, so it is helpful to learn how to think about them.
In this chapter we looked at momentum, which is a way of quantifying motion. The momentum of a system is defined by ptot = m1v1 + m2v2 + …. Momentum has a kind of permanence, so an isolated system has a constant momentum in time, and we can change the momentum of a system only by applying an unbalanced external force. This is called conservation of momentum.
In any problem involving a collision, especially if there is crunching or sticking, momentum is likely to be a key concept. In these problems the external forces are negligible if the collision is brief, and the internal forces are very complicated. The equation for conservation of momentum can quickly lead to answers.

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