# Spacetime Diagram

1. Apr 29, 2010

### Lissajoux

1. The problem statement, all variables and given/known data

To draw a carefully labelled spacetime diagram for the following situation:

The addition of velocities as viewed from a ship, for the case of a missile launched at $v_{mi}=+0.80c$ relative to the ship, itself travelling at $$v_{s}=+0.60c[/itex] relative to the Earth. Take the launch time to be at $t=0$ as the spaceship passes the Earth. Use the diagram to discuss the velocity of the missile relative to Earth according to observers on Earth and on the spaceship. 2. Relevant equations Within the problem statement and solution attempt. 3. The attempt at a solution This is what I have for the spacetime diagram (see the included image) but I'm not sure if the scales and labels are correct, I just made the 3,4,5 values as that was convenient numbers that seem to work from some quick calculations, I think I can just do that. http://img188.imageshack.us/img188/3645/spacetimediag1.jpg [Broken] Hopefully that diagram is correct. Now this is the bit I'm more unsure of - finding the velocity of the missile relative to the Earth and to the ship. This is what I have so far.. Can state that the ship leaves Earth at speed $v_{sh}=+0.6c$$ in frame [itex]F$. From the ship viewpoint in $F'$ the Earth leaves the ship at speed $-v_{sh}=-0.6c$. The missile fired from the ship moves at speed $u'$ in frame $F'$ at $O$.

The relative speed of the Earth and the missile is:

$$\frac{\Delta x'}{t'}=\frac{RQ}{OP}=u'+v$$

.. as seen from the ship in frame $F'$

This is allowed to be $> c$ since this is not the speed of the missile relative to the Earth measured on Earth.

Then in regards to the missile speed $u$ measured in $F$.. the relative speed in $F$ is:

$$u=\frac{\Delta x}{t}=\frac{RS}{OR}$$

Need to deduce these results via geometry and algebraic calculus, i.e. using that diagram and what values are known, but I can't figure out how to do all that.

If I can get a bit of advice that would be great to get me going with this question

Last edited by a moderator: May 4, 2017