Spacetime Diagram: Velocity Relative to Earth/Ship

[Lissajoux]

Homework Statement

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 [itex]v_{mi}=+0.80c[/itex] relative to the ship, itself traveling at [tex]v_{s}=+0.60c[/itex] relative to the Earth.

Take the launch time to be at [itex]t=0[/itex] 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.

Homework Equations

Within the problem statement and solution attempt.

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

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 [itex]v_{sh}=+0.6c[/tex] in frame [itex]F[/itex]. From the ship viewpoint in [itex]F'[/itex] the Earth leaves the ship at speed [itex]-v_{sh}=-0.6c[/itex]. The missile fired from the ship moves at speed [itex]u'[/itex] in frame [itex]F'[/itex] at [itex]O[/itex].

The relative speed of the Earth and the missile is:

[tex]\frac{\Delta x'}{t'}=\frac{RQ}{OP}=u'+v[/tex]

.. as seen from the ship in frame [itex]F'[/itex]

This is allowed to be [itex]> c[/itex] since this is not the speed of the missile relative to the Earth measured on Earth.

Then in regards to the missile speed [itex]u[/itex] measured in [itex]F[/itex].. the relative speed in [itex]F[/itex] is:

[tex]u=\frac{\Delta x}{t}=\frac{RS}{OR}[/tex]

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

 

[mark726]

Your spacetime diagram looks correct in terms of the scales and labels. However, for the problem at hand, it would be more helpful to have a spacetime diagram in the frame of reference of the Earth, instead of the spaceship. This will allow you to better analyze the situation and calculate the velocities of the missile relative to the Earth and the spaceship.

To draw the spacetime diagram in the frame of reference of the Earth, you can use the Lorentz transformation equations. These equations will allow you to transform the coordinates and time measurements from the frame of reference of the spaceship to the frame of reference of the Earth.

Once you have the spacetime diagram in the frame of reference of the Earth, you can use the geometry and algebraic calculus to determine the velocities of the missile relative to the Earth and the spaceship.

Remember that the velocity of an object is the change in position over time. So, to determine the velocity of the missile relative to the Earth, you need to calculate the change in position of the missile as observed by an observer on the Earth, and then divide it by the time measured by the same observer.

I hope this helps. Good luck with your calculations!