Special Relativity — practice questions

A muon's internal clock measures its lifetime as $t_0 = 2.20\ \mu\text{s}$. A lab observer sees the muon moving at $v = 0.60c$.

Calculate the Lorentz factor $\gamma$ and the lifetime measured in the lab.

A spacecraft clock runs for $t_0 = 5.0$ s as measured on board. An Earth-based observer measures the same interval as $t = 8.33$ s.

Calculate the speed of the spacecraft relative to Earth, as a fraction of $c$.

Two observers disagree about how long an event lasts, yet both agree on the speed of light.

Explain, starting from Einstein's postulates, why this is not a contradiction, and why proper time is always the shortest time measured for a given pair of events.

A metre stick has a proper length of $1.00$ m. It moves past an observer at $v = 0.60c$, so $\gamma = 1.25$.

Calculate the length the observer measures.

A spacecraft's proper length is $80.0$ m. An Earth observer measures its length as $48.0$ m as it flies past.

Calculate the spacecraft's speed as a fraction of $c$.

A cylindrical spacecraft travels horizontally at high speed. Explain what an observer on Earth measures for its length and its diameter, and explain why these two dimensions are affected differently.

A muon has a proper lifetime of $2.2\ \mu\text{s}$ and travels at $v = 0.98c$, giving $\gamma = 5.03$.

Calculate the muon's lifetime as measured by an observer on Earth.

A muon travels at $v = 0.98c$ ($\gamma = 5.03$) toward the ground, $15\,000$ m away as measured from Earth.

Calculate the distance to the ground as measured in the muon's own frame, and hence show it is plausible for the muon to survive the trip given its proper lifetime of $2.2\ \mu\text{s}$.

Explain, using both the Earth frame and the muon's frame, why more muons reach the ground than classical physics predicts, and explain why the two explanations must give the same result.

State the equivalence principle, and state one observable consequence of it.

Explain why the equivalence principle implies that light must be bent by gravity.

Compare the time dilation predicted by special relativity with that predicted by general relativity, and explain how the GPS system demonstrates the connection between them.