# I Help me understand relative speeds

#### some bloke

Summary
the laws of physics dictate that nothing can exceed C, but if 2 ships travel away from one another at 0.6C, they would perceive each other as travelling faster than C (1.2C).
at what point does something class as travelling this fast?
This came to me in a blur of confusion whilst reading another thread about a giant wheel spinning, and it's made me confused. I'm hoping that this is something which someone can explain to me in a way I'll understand.

Summary of my knowledge:
1: the speed of light is the same in all directions, no matter how fast you are travelling, as discovered in the "Ethereal Wind" experiment.
2: no physical object can reach the speed of light

But what I'm confused about is what speed of light?

For example:
2 rockets leaving the earth, in opposite directions. The rockets accelerate to travel at 0.6C, relative to the earth.
Rocket 1 looks at rocket 2 and sees it moving away at 1.2C, and sees itself as stationary. Rocket 2 observes the same about rocket 1.

The same experiment works if Rocket 1 and Rocket 2 are passing each other at 0.6C, they will see themselves as stationary and the other as flying past at 1.2C.

Now imagine they are orbiting a point, 180° offset from one another, in the same direction (so they are always travelling in opposite linear directions from one another). This imitates a wheel, with the point as a hub and the ships as the outer edge. They travel at 0.6C, see one another as travelling at 1.2C but also at 0, as they are not moving compared to one another.

I don't really understand how these interactions play out. can someone help me?

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#### vanhees71

Gold Member
By definition, the speed of rocket 1 relative to rocket 2 is the speed of rocket 1 as measured in the restframe (SIC!) of rocket 2.

#### PeroK

Homework Helper
Gold Member
2018 Award
Summary: the laws of physics dictate that nothing can exceed C, but if 2 ships travel away from one another at 0.6C, they would perceive each other as travelling faster than C (1.2C).
at what point does something class as travelling this fast?

This came to me in a blur of confusion whilst reading another thread about a giant wheel spinning, and it's made me confused. I'm hoping that this is something which someone can explain to me in a way I'll understand.

Summary of my knowledge:
1: the speed of light is the same in all directions, no matter how fast you are travelling, as discovered in the "Ethereal Wind" experiment.
2: no physical object can reach the speed of light

But what I'm confused about is what speed of light?

For example:
2 rockets leaving the earth, in opposite directions. The rockets accelerate to travel at 0.6C, relative to the earth.
Rocket 1 looks at rocket 2 and sees it moving away at 1.2C, and sees itself as stationary. Rocket 2 observes the same about rocket 1.

The same experiment works if Rocket 1 and Rocket 2 are passing each other at 0.6C, they will see themselves as stationary and the other as flying past at 1.2C.

Now imagine they are orbiting a point, 180° offset from one another, in the same direction (so they are always travelling in opposite linear directions from one another). This imitates a wheel, with the point as a hub and the ships as the outer edge. They travel at 0.6C, see one another as travelling at 1.2C but also at 0, as they are not moving compared to one another.

I don't really understand how these interactions play out. can someone help me?
Velocities do not simply add in the way that you imagine. In the case of two rockets moving in opposite directions relative to the Earth, say, the speed of one rocket (as measured in the reference frame of the other is):

$\frac{0.6 + 0.6}{1 + (0.6)^2}c \approx 0.88c$

#### Ibix

2 rockets leaving the earth, in opposite directions. The rockets accelerate to travel at 0.6C, relative to the earth.
Rocket 1 looks at rocket 2 and sees it moving away at 1.2C, and sees itself as stationary. Rocket 2 observes the same about rocket 1.
As @PeroK says, velocities don't add linearly. A velocity of $u$ and a velocity of $v$ combine to make a velocity of $(u+v)/(1+uv/c^2)$. Note that this is almost indistinguishable from $u+v$ if both $u$ and $v$ are very much less than $c$, which is why you don't notice this at every day speeds.
Now imagine they are orbiting a point, 180° offset from one another, in the same direction (so they are always travelling in opposite linear directions from one another). This imitates a wheel, with the point as a hub and the ships as the outer edge. They travel at 0.6C, see one another as travelling at 1.2C but also at 0, as they are not moving compared to one another.
If you have switched to a frame of reference where the two ships are not moving, this is a rotating frame of reference. This involves a fairly major change in what you mean by "velocity", and the speed of light is neither constant nor isotropic, nor is $c$ a maximum speed limit in such a frame. This doesn't mean anything particularly exotic - simply that the definitions of "velocity" one has to use in a rotating frame are complicated.

My advice is not to worry too much about rotating reference frames until you have a good understanding of inertial frames.

Last edited:

#### metastable

Summary: the laws of physics dictate that nothing can exceed C, but if 2 ships travel away from one another at 0.6C, they would perceive each other as travelling faster than C (1.2C).
at what point does something class as travelling this fast?

I don't really understand how these interactions play out. can someone help me?
Some may see the scenario outlined in this post as over-complicated, but it helped me understand the concepts a bit better:

#### David Lewis

… 2 rockets leaving the earth, in opposite directions. The rockets accelerate to travel at 0.6C, relative to the earth.
For you on Earth, with each second that passes, the distance between the rockets increases by 1.2 light-seconds.

#### jbriggs444

Homework Helper
For you on Earth, with each second that passes, the distance between the rockets increases by 1.2 light-seconds.
Indeed. Some things are still just that simple.

Of course this is not the "relative velocity" of any one thing in the rest frame of another. Instead is it a "closing velocity". Further, it is expressed as a ratio of so much Earth-relative distance in so much Earth-relative time.

When translated to the rest frame of either rocket we need to transform coordinates so that we are talking in terms of rocket-relative distances and rocket-relative times to obtain a rocket-relative velocity. Time dilation, length contraction and the relativity of simultaneity all have roles to play in the transformation. The result will be the velocity of one physical thing in the rest frame of another and will have a magnitude less than c.

"Help me understand relative speeds"

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