Relative Velocity of Two Objects Moving at Different Speeds

[Tranceform]

Homework Statement

An intergalactic cruiser is approaching a hostile spacecraft . Both vehicles move at a constant velocity. The velocity of the cruiser relative to the spacecraft is v_CS=+0.7c, the direction to the right being the positive direction. The cruiser fires a beam of laser light at the hostile renegades. The velocity of the laser beam relative to the cruiser is v_LC=+c.

a) What is the velocity v_LS of the laser beam relative to the renegades’ spacecraft ?

b) What is the velocity v at which the renegades see the laser beam move away from the cruiser?

Homework Equations

The Speed-of-Light Postulate: The speed of light in a vacuum, measured in any inertial reference frame, always has the same value of c, no matter how fast the source of light and the observer are moving relative to each other.

The Attempt at a Solution

In both cases, it is asked how the speed of light is percieved. Question a) makes sense, as it's asking for the speed of light relative to an inertial reference frame. According to The Speed-of-Light Postulate of special relativity, light traveling in vacuum perceived in an inertial reference frame always has the value of c so the answer is v_LS=+c. What I'm wondering about is question b). In order to answer this question I need to understand the following:

How is question b) different from question a)? In question b) it is also being asked how the speed of light is perceived viewed from an inertial reference frame, so according to the abovementioned postulate it should also be +c. But that is not the correct answer. Why is it not that?

 

[Simon Bridge]

Think of it like this, if I am going at 2m/s wrt you, and I throw a ball at 5m/s towards you (also wrt you), how fast do you see the ball leaving my hand?

 

[Tranceform]

Think of it like this, if I am going at 2m/s wrt you, and I throw a ball at 5m/s towards you (also wrt you), how fast do you see the ball leaving my hand?

Is this analogy in reference to question a) or b)? I thought this analogy didn't apply for light? Suppose for example that you instead of throwing a ball shined a flashlight, the speed of light postulate says that I wouldn't see this light as c+2m/s even though you are moving but I would still see it as having speed c?

 

[Simon Bridge]

Refers to part (b). All the laws of physics have to work normally in an inertial reference frame, as long as all measurements are taken by the same inertial observer.

Suppose for example that you instead of throwing a ball shined a flashlight, the speed of light postulate says that I wouldn't see this light as c+2m/s even though you are moving but I would still see it as having speed c?

You need to read my analogy more carefully.
You would not see the ball leave my hand at 2+5=7m/s either.
You would still see the ball move wrt you at 5m/s.
I am moving wrt you at 2m/s.
The gap between me and the ball is increasing.
So how fast is the gap between me and the ball increasing?
That is how fast the ball leaves my hand, as measured by you.

Lets say I am using a flashlight, and I use it to fire a pulse of light at you.
So I am moving towards you at speed 2m/s
The light pulse is moving towards you at speed c.
The gap between me and the light pulse is increasing - how fast?

That is what part (b) is asking.

 

[Tranceform]

Thanks for clarifying!

 

[Simon Bridge]

No worries - it's very common. The key to relativity problems, all kinds, is to be very careful about who is doing the observing/measuring etc. What are the measurements being made with respect to? Sometimes the clue is only in one or two words.

Try this:

If Alice travels to the left, according to Oscar, at 0.9c, and Bob travels to the right, according to Oscar, at 0.9c, how fast is the distance between Alice and Bob increasing (according to Oscar)?