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Special relativities

A space ship, at rest in a certain reference frame S, is given a speed increment of
0.46c (call this boost 1). Relative to its new rest frame, the spaceship is given a
further 0.46c increment 12s later (as measured in its new rest frame; call this boost
2). This process is continued indefinitely, at 12s intervals, as measured in the rest
frame of the spaceship. Assume that each boost takes a negligibly short time
compared with 12s.

1. derive an equation that gives the velocity of the ship after any boost n (n==1,2,3,4....)

2. show the value of the "gamma function for boosts 1,.2,3,...10

gamma = [1-(v^2/c^2)]^.5......c= speed of light in meters/sec

Solution Preview

Hi C!

Here is the solution. It actually shows a nifty way to add velocities in many ref frames that move with different velocities with respect to each other).

Note that I used the notation where c=1, so teh velocities are all in units of c and when I write the speed v=0.75 it means that the speed is actually 0.75c

Let's examine the one dimensional Lorentz transformation in a matrix form:

Where v is expressed as a fraction of c=1 and:

Since we can write:

So:

Hence:

Note that the definition of  is simply now:

Now assume that at a moving frame S' with relative velocity v1 to frame Sand another reference frame S" moving with ...

Solution Summary

A space ship, at rest in a certain reference frame S, is given a speed increment of
0.46c (call this boost 1). Relative to its new rest frame, the spaceship is given a
further 0.46c increment 12s later (as measured in its new rest frame; call this boost
2). This process is continued indefinitely, at 12s intervals, as measured in the rest
frame of the spaceship. Assume that each boost takes a negligibly short time
compared with 12s.

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