What is proper time?

State the time dilation formula.

Which interval is shorter: proper time \(\Delta\tau\) or dilated time \(\Delta t\)?

A moving clock has proper tick interval \(2.00\,\mathrm{ns}\). If \(\gamma=3.00\), what interval is measured in the lab?

A particle moves at \(0.800c\). Find \(\gamma\).

A spacecraft clock measures \(10.0\,\mathrm s\) between two events on board. The spacecraft moves with \(\gamma=1.25\) relative to Earth. What time interval does Earth measure?

A muon has proper lifetime \(2.20\,\mu\mathrm s\) and moves with \(\gamma=10.0\). Find its mean lifetime in the laboratory.

A clock moving at \(0.600c\) measures \(4.00\,\mathrm s\). Find the interval measured by a stationary lab observer.

A lab measures a moving particle lifetime of \(8.00\,\mu\mathrm s\). If \(\gamma=4.00\), find the proper lifetime.

A muon moves at \(0.995c\) and has proper lifetime \(2.20\,\mu\mathrm s\). Estimate the distance it travels in the lab during one mean lifetime.

At what speed does a moving clock run slow by a factor of \(2.00\) compared with the lab?

An astronaut ages \(3.00\) years during a trip. Earth measures \(5.00\) years for the same departure-arrival events. Find \(\gamma\) and the ship speed.

A particle travels \(1.50\,\mathrm{km}\) in the lab at \(0.900c\). Find the proper time elapsed on the particle.

Why is the proper time between two ticks of a moving clock measured in the clock's own frame?

A clock's moving-frame interval is \(12.0\,\mathrm{ns}\), and its proper interval is \(5.00\,\mathrm{ns}\). Is this possible? If so, find the speed.

Derive time dilation from the Lorentz time transformation for two events at the same place in \(S'\).

A student uses \(\Delta t=\Delta\tau/\gamma\) and gets a lab lifetime smaller than the particle lifetime. Identify the error.

Explain how muons produced high in the atmosphere can reach Earth's surface using time dilation.

Why is it incomplete to say simply "moving clocks run slow" without naming a frame?

Show that time dilation disappears in the Newtonian limit.