Problem 3

A spacecraft moves at constant speed

v

relative to Earth. Its onboard clock measures proper time

\Delta\tau=2.0

years between two events occurring at the same spacecraft location. Given light speed

c

and

v=0.8c

, find the time between the events according to Earth clocks and the distance traveled by the spacecraft. Model each stage with the appropriate conservation law or equation of motion; quantities stated as constant remain fixed during that stage. (a) Establish the governing physical relation for each stage. (b) Determine the requested observables from the stated initial conditions and constraints. (c) Check signs, dimensions, and the physical limiting behavior of the results.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\gamma=\frac{1}{\sqrt{1-0.8^2}}=\frac53

problems.proof.analysis

Substitution of the given speed gives $\gamma=5/3$ . This follows from the chosen physical model and the stated initial conditions; the sign convention must remain consistent throughout. A dimensional check catches algebraic errors before the result is interpreted.

problems.proof.pitfall. Do not mix sign conventions or omit a system constraint; check the sign and units before interpreting the result.