Suppose you want to move an empty paper clip box by shooting it with a toy dart gun. Why would you want to do this? Don’t worry about that – this is my example and I am sticking with it. Should you shoot a dart that sticks to the box or should you shoot one that bounces off? I made a video of this exact situation. Note: you could obviously come up with other objects to do this with, but I always like to use more normal stuff.

In case it wasn’t clear, the first dart bounced back and made the box go much faster (and farther) than the dart that stuck (inside) the box. The usual question is: which dart had a greater change in momentum? You could also look at this in terms of impulse. First, the momentum principle:

In this form, it says that the product of net force and time the force is acting on an object is the change in momentum of that object. For this case, there is a collision. So, the important points for a collision are that the forces between the two colliding objects have are equal and opposite and that they last for the same amount of time. This means the the change in momentum of one object is the negative of the change in momentum of the other object.

With that idea, you can see already which has a larger change in momentum. When the dart bounced off (instead of sticking into) the box, the box had a higher speed (and went farther). So since the change in momentum of the box was larger in this case, so was the change in momentum of the dart. Time for another picture. Here is the dart bouncing off the box.

If the dart bounces back at a little bit lower speed, the change in momentum (in the x-direction) will be:

This is where many people make the mistake in saying the change in momentum is 2 kg*m/s. Ah ha! That is the change in the magnitude of the momentum, not the change in the momentum. You see, they are different.

Since the change in momentum for the dart is the same (but opposite direction) as the change in momentum of the box, it increases in momentum by 8 kg*m/s (and it started at zero). Now, here is a diagram for the case where the dart sticks. (I will assume that it starts with the same initial momentum)

So, in this case, the change in momentum of the dart is: (in the x-direction)

This means that the box must have a change in momentum of 2 kg*m/s (and thus is slower than the case where the dart bounced off).