**pre reqs:** [Vectors and Vector Addition](http://scienceblogs.com/dotphysics/2008/09/basics-vectors-and-vector-addition.php)

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://scienceblogs.com/dotphysics/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://scienceblogs.com/dotphysics/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).

Clearly this works for the simple case above, but actually it works no matter which direction as long as the equation remains as a **vector** equation. In general, with two reference frames (Say A and B) then you (or I) can say:

![Screenshot 17](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/screenshot-17.jpg)

The most important thing is that these are vectors and must be treated as such. If you treat these vectors as scalars, you will likely get the problem wrong.

Ok. Fine, that makes some sense – but these darn physics problems are killing me (or you). How about an example, everyone loves those. However, if you don’t feel comfortable with vectors, go look [at my introduction to vectors](http://scienceblogs.com/dotphysics/2008/09/basics-vectors-and-vector-addition/).

**Problem** Suppose you have a boat that travels at a speed of 2 m/s on the water. This boat is to cross a river that is 500 meters wide and has a speed (of the water) of 0.5 m/s. What angle should you aim your boat so that it travels straight across the river (without going downstream at all).

Here is a picture:

![Screenshot 19](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/screenshot-19.jpg)

What is given in the problem? There is the magnitude of the velocity of the boat with respect to the water, the velocity of the water with respect to the ground. There is one other thing that the problem gives. The velocity of the boat with respect to the ground is only in the y-direction. The goal of the problem is to find the angle ? that the boat has to aim. Here is what is given (re-written):

![Screenshot 20](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/screenshot-20.jpg)

Actually, there is one other piece of information that is important. The velocity of the boat with respect to the ground is ONLY in the y direction. I can write this as:

![Screenshot 21](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/screenshot-21.jpg)

Now, let me put these velocities together:

![Screenshot 22](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/screenshot-22.jpg)

Where this is a vector addition equation. To add vectors, I can just add the x-components and then just add the y-components. In this case, I can JUST look at the x-direction:

![Screenshot 23](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/screenshot-23.jpg)

Note that the (m/s) units cancel. Solving for sin(?):

![Screenshot 25](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/screenshot-25.jpg)

So, there you have it. Let me recap what is important:

- Start with the relative velocity equation
- Write down the velocities you know (as vectors)
- Treat the velocities as vectors

See. That is not too bad, is it?

**Final Note:** This is known as Galilean relativity. It works when the velocities of the frames and objects are much less than the velocity of light. (example: a jet going at twice the speed of sound is way slower than light). If the objects are moving close to the speed of light, this stuff does not work.