# Relative Velocities (AP)

Suppose that Curly challenges Moe and Larry to a race. Their starting positions are shown in the diagram below.

One second later their positions are:

Notice that Curly didn't actually start, and Moe carries his measuring stick along with him.

Now, Curly notes that Larry started at position x_{o} = 0 m and one second later he was at position x = 6 m, so Curly measures Larry's velocity as 6 m/s. Let's write that as v_{LC} = 6 m/s - meaning "velocity of Larry measured by Curly is 6 m/s."

Curly also notes that Moe started at position x_{o} = 0 m and one second later he was at position x = 4 m, so Curly measures Moe's velocity as 4 m/s. Let's write that as v_{MC} = 4 m/s - meaning "velocity of Moe measured by Curly is 4 m/s."

Moe sees that Larry started at position x_{o} = 0 m and one second later he was at position x = 2 m (on Moe's measuring scale, of course), so Moe measures Larry's velocity as 2 m/s. Let's write that as v_{LM} = 2 m/s - meaning "velocity of Larry measured by Moe is 2 m/s."

Curly and Moe measure different velocities for Larry. Sure, velocities are relative. But how do these velocities fit together? Well,

v_{LC} = v_{LM} + v_{MC}

This is often called Galilean or classical relativity. When you think about it, it makes perfect sense. The only problem with it is that it doesn't *always* work.

## Relativistic Velocities

The Galilean relativity equation works just fine for "normal" velocities, like running Stooges or even jet planes, but not so well when velocities approach the speed of light. Suppose that Curly measures Moe's velocity as eight-tenths of the speed of light (v_{MC} = 0.8c), and Moe measures Larry's velocity as eight-tenths of the speed of light (v_{LM} = 0.8c). Using the classical relativity equation above, you get:

v_{LC} = v_{LM} + v_{MC} = 0.8c + 0.8c = 1.6c (!!??)

In other words, this predicts that Curly will measure Larry's velocity as 1.6 times the speed of light, which is forbidden by Einstein's Special Theory of Relativity, which says, among other things, that the speed of light is the ultimate speed limit in the Universe. In Special Relativity the classical equation is modified to be:

Using Special Relativity, we get:

In other words, Curly would measure Larry's velocity as 98% of the speed of light. Can this be correct? Yes, it can. For one thing, it has been very well-confirmed by experiment - not with Stooges, but with sub-atomic particles!

*last update October 13, 2008 by JL Stanbrough*