Calculating Time and Distance in Lorentz Transforms: A Homework Example

[mattst88]

Homework Statement

At x = x' = 0 and t = t' = 0, a clock ticks on a fast spaceship (gamma = 100). The captain of the ship heads it tick again 1.0 s later. Where and when do we (the stationary observers) measure the second tick to occur?

Homework Equations

$$t = \frac{t'}{\gamma}$$
$$x' = \gamma(x - vt)$$
$$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

The Attempt at a Solution

First, I solve for velocity.
$$v = c\sqrt{1 - \frac{1}{100}^2} = 0.9995c$$

Next, solve for t when t' = 1.
$$t = \frac{t'}{\gamma} = \frac{1}{100} sec$$

Finally, solve for x'.
$$x' = \gamma(x - vt) = 100(0 - (0.9995c)(\frac{1}{100})) = 0.9995c$$

Have I answered this correctly? Thanks.

 

[BerryBoy]

The spaceship is defined as the moving object, so proper time is measured on the Earth. Time appears to be going slower for the spaceship. So we could write that:

$$t_{earth} = \gamma t_{spaceship}$$

Sam