Relative Velocities A & B: 5MPH Each

[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.

 

[Doc Al]

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

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.

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.

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"

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

 

[Dale]

Hi ssope, welcome to PF.

The correct formula for adding velocities is called the http://en.wikipedia.org/wiki/Velocity-addition_formula" :

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

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

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

 

[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.

 

[ssope]

Thank you all very much for answering my question.