# An example of two collisions

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).

February 4, 2010

Good Example! I teach conceptual physics classes in high school and enjoy clear and easy to do experiments that my students can do at their desks. Thank you for the video.

2. #2 Dylan Przygocki
February 5, 2010

Hi,
I had a question about a completely unrelated field to this thread, but still a physics question I thought you may be able to help me out with.
I’m looking for information on sound technology, either present, past, or in development, that is designed to control the flow of water. A friend told me about a rather fanciful acoustic concept in which sound could possibly be used to create a vacuous space within water. I was wondering about the possibilities of this and whether this concept matches anything currently being studied.
Thanks, and sorry if this is totally the wrong place for this comment.
Dylan

3. #3 Rhett Allain
February 5, 2010

@Dylan,

Sorry, but I really don’t know anything about water controlled with acoustics. If I find something, I will let you know.

4. #4 Dan
February 6, 2010

How did you get it to stick the second time? It looked like the same dart. I might like to try this with my high school physics class.

5. #5 Rhett Allain
February 6, 2010

@Dan,

There is a hole on one side of the box. The dart shoots into and sticks inside the box. I had to play around with the size of the hole a little to get it to stick.

If you have a good dart, you can use the suction cup to stick it – I couldn’t get that to work.

6. #6 Dan
February 7, 2010

Thanks, I am going to try this.

7. #7 Allison
February 7, 2010

You inspired me to go out and buy a nerf gun for my physics classes. Thanks!

8. #8 Rhett Allain
February 8, 2010

@Allison,

Everyone needs a nerf gun. I will try to come up a list of nerf-friendly physics experiments.