# Relative velocities in relativity

• tfast
In summary, the conversation discusses how observers on two spacecrafts traveling towards each other at different velocities would perceive each other's movements and how this is affected by relativistic effects. The correct formula for adding velocities is mentioned and it is noted that the speed of any object relative to another will never exceed the speed of light.
tfast
Hi--

I'm just trying to clean out some physics cobwebs in my head, as I was never much up on relativity; thanks in advance for accomodating me. Here's my question:

Imagine two spacecraft widely separated, but traveling toward each other at significant fractions of c (observed relative to the same reference object, say a sun). What would observers on each spacecraft see in the following situations:

a) each ship travels at v < 0.5c relative to the reference object
b) each ship travels at v = 0.5c relative to the reference object
c) each ship travels at v > 0.5c relative to the reference object

My naive assumption in the case of (a) would be that observers on either ship would see the other ship approaching at 2(v), with light from the other ship blue-shifted in proportion to that speed. However, this doesn't seem to take into account dilation effects, so I'm not really sure of this answer.

So, is the answer above correct, and furthermore, what happens when the ships' relative velocities sum to >= c in cases (b) and (c)? Finally, does anything change when the ships approach one another with velocities different from one other?

Thanks!

Originally posted by tfast
Hi--

However, this doesn't seem to take into account dilation effects, so I'm not really sure of this answer.

So, is the answer above correct, and furthermore, what happens when the ships' relative velocities sum to >= c in cases (b) and (c)?

Thanks!

w = u+v but by

w = (u+v)/(1+uv/c²)

This is the correct formula for all additions of velocity. Note that when uv<<c (As in everyday experence), The formula gives an answer nearly identical with u+v, Which is why we still tend to use w = u+v for low velocity situations.

Just come extra notes ...
Note that as u->c and v->c the equation will reduce to :
w = u+v/(1+c2/c2)
w = u+v/2
so if u=~v
w =~ u =~ v
And this is why the speed of any object (relative to any other object) will never pass the speed of light.

Also note this equation is called "Lorentz Transformations"

Cheers all, thanks for the excellent answers!

## 1. What is the concept of relative velocities in relativity?

Relative velocities in relativity refer to the way that the speed of an object is perceived and measured by different observers who are in relative motion to each other.

## 2. How does Einstein's theory of relativity explain relative velocities?

Einstein's theory of relativity explains relative velocities by stating that the laws of physics are the same for all observers in uniform motion, regardless of their relative velocities. This means that the speed of light is constant for all observers, regardless of their relative motion.

## 3. Can relative velocities exceed the speed of light?

No, according to Einstein's theory of relativity, the speed of light is the maximum speed in the universe. Therefore, relative velocities cannot exceed the speed of light.

## 4. How does time dilation affect relative velocities?

Time dilation, which is a consequence of Einstein's theory of relativity, states that time passes slower for objects in motion compared to stationary objects. This means that for observers in relative motion, time will appear to pass at different rates, which can affect how they perceive and measure relative velocities.

## 5. Are relative velocities only applicable to objects in space?

No, relative velocities are applicable to all objects, regardless of their location. This includes objects on Earth as well as those in space. The concept of relative velocities is important in understanding how objects move and interact in the universe.

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