# Homework Help: Relativistic speeds within a relativistic frame of reference

1. Jul 1, 2014

### xuran

Astronauts on a long space journey are playing golf inside their spaceship, which is travelling away from the Earth with speed 0.6c. One of the astronauts hits a drive exactly along the length of the spaceship (in its direction of travel) at speed 0.1c in the frame of the spaceship.

What is the speed of the gold ball as observed from Earth?
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So to the people on earth, the spacecraft itself is obviously going to appear to travel at 0.6c.
However since the spacecraft is travelling at relativistic speeds, I thought the speed of the golf ball would not appear to travel at 0.1c to an observer on earth. Instead, length would contract in the direction of motion so it would appear to cover less distance when observed from earth.
So the ratio in which the speed should contract would be given by:
√(1-(0.6)2) = 0.8 as this is ratio of how much length contracts.
And so due to length contraction, the speed of the golf ball would appear to travel at 0.8 x 0.1c. So the ball would go at 0.08c

Just adding this to 0.6, the ball would appear to travel at 0.68c. I wasn't to sure if this was right, considering I've never come across a question that involved an object travelling at relativistic speeds within frame of reference that appears to also be moving at a relativistic speed. On top of this, the answer is apparently wrong.

The answer says 0.66c but offers no working out. Are the answers wrong or am I missing something?

2. Jul 1, 2014

### Orodruin

Staff Emeritus
Your argumentation does not work. The time in the different frames behave differently and it is not a simple question of adding two numbers together. You have to use relativistic addition of velocities. If you have a course book, it should be described in it.

3. Jul 1, 2014

### xuran

All my textbooks (three) only give the equations for time dilation, length contraction and mass dilation. We are given a very basic rundown of what special relativity is that pretty much just involves simultaneity and the above three changes.
With just this would it be expected to know how to add relativistic velocities? Is there a way to derive out an equation to add relativistic velocities considering my limited knowledge? (I googled it, plugged the numbers in and got the correct answer).

It is possible this test has asked a question out of the scope of the course.

4. Jul 1, 2014

### dauto

Yes of course it is possible to derive the formula. That's how the formula you found on line is obtained. But typically that formula would be derived in the text book. If your books do not include that formula than I would agree that this problem is out of the scope of those books. Have you seen anything about Lorentz transformation?