# Relative Velocities

1. Dec 19, 2009

### ssope

A approaches B at 5 MPH
B approaches A at 5 MPH

I am wondering why at very fast speeds, the error would become quite large if you were to say that A and B's relative velocity is equal to 10.

2. Dec 19, 2009

### Staff: Mentor

I assume you mean something like this:
A moves towards B at a speed of 5 mph with respect to some frame C.
B moves towards A at a speed of 5 mph with respect to some frame C.

It's a conclusion of special relativity that velocities do not add simply as V1 + V2. Read all about it: http://math.ucr.edu/home/baez/physics/Relativity/SR/velocity.html" [Broken]

(Edit: I forgot to add the punchline, that the difference becomes marked when speeds approach light speeds. DaleSpam got it.)

Last edited by a moderator: May 4, 2017
3. Dec 19, 2009

### Staff: Mentor

Hi ssope, welcome to PF.

$$\frac{v_1+v_2}{\frac{v_1 v_2}{c^2}+1}$$

$$\frac{5+5}{\frac{5 \times 5}{(6.7 \times 10^8)^2}+1} = 9.9999999999999994 \, mph$$

For such low velocities the difference between the real formula and the approximation is undetectable, less than 1 micrometer/century.

Last edited by a moderator: May 4, 2017
4. Dec 19, 2009

### sweet springs

Hi,
A approaches C at 5 MPH
C approaches B at 5 MPH
Then
For C: A and B's relative velocity of approach equal to 10.
For A: the velocity of B is less than 10.
For B: the velocity of A is less than 10.
Regards.

Last edited: Dec 19, 2009
5. Dec 19, 2009

### ssope

Thank you all very much for answering my question.