**pre reqs:** [Vectors and Vector Addition](http://blog.dotphys.net/2008/09/basics-vectors-and-vector-addition/)

This was sent in as a request. I try to please, so here it is. The topic is something that comes up in introductory physics – although I am not sure why. There are many more important things to worry about. Let me start with an example. Suppose you are on a train that is moving 10 m/s to the right and you throw a ball at 5 m/s to the right. How fast would someone on the ground see this ball? You can likely come up with an answer of 15 m/s – that wasn’t so hard right? But let me draw a picture of this situation:

![Screenshot 15](http://blog.dotphys.net/wp-content/uploads/2008/09/screenshot-15.jpg)

The important thing is: If the velocity of the ball is 5 m/s, that is the velocity with respect to what? In the diagram, I listed the velocity of the ball as *v_{ball-train}* this indicates it is with respect to the train. There are three velocities in this example.

- The velocity of the ball with respect to the train
- The velocity of the train with respect to the ground
- The velocity of the ball with respect to the ground

These three velocities are related by the following:

![relative v](http://blog.dotphys.net/wp-content/uploads/2008/09/relative-v.jpg)

**note**: The way I always remember this is to arrange it so that the frames match up on the left side. That is to say v(a-b) + v(b-c) – you can think of this as the “b’s” canceling and giving v(a-c).