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Time Dilation

A public special-relativity question showing how to identify the proper time and apply the Lorentz factor to a moving clock.

Question

A starship travels at v=0.6cv = 0.6c relative to Earth. The crew measures 8.0 s8.0\,\text{s} between two ticks of a clock on board. How much time elapses between those same two ticks as measured by Earth observers?

  1. a.12.5 s12.5\,\text{s}
  2. b.10 s10\,\text{s}
  3. c.8.0 s8.0\,\text{s}
  4. d.6.4 s6.4\,\text{s}

Correct answer

B. 10 s10\,\text{s}

Full reasoning

  1. 1The crew's 8.0 s8.0\,\text{s} is the proper time Δτ\Delta\tau, because both ticks occur at the clock's own location in the ship frame.
  2. 2Compute the Lorentz factor: γ=1/1−(0.6)2=1/0.64=1/0.8=1.25\gamma = 1/\sqrt{1 - (0.6)^2} = 1/\sqrt{0.64} = 1/0.8 = 1.25.
  3. 3Earth observers measure the dilated interval Δt=γΔτ=1.25×8.0=10 s\Delta t = \gamma\Delta\tau = 1.25 \times 8.0 = 10\,\text{s}.

Why each choice is right or wrong

Choice A

This uses γ=1/(1−v2/c2)=1/0.64=1.5625\gamma = 1/(1 - v^2/c^2) = 1/0.64 = 1.5625 without taking the square root in the denominator.

Choice B

Correct. γ=1/1−0.36=1/0.8=1.25\gamma = 1/\sqrt{1 - 0.36} = 1/0.8 = 1.25, and the dilated interval is 1.25×8.0=10 s1.25 \times 8.0 = 10\,\text{s}.

Choice C

This assumes both frames share a universal time; in special relativity the moving clock's interval is dilated for Earth observers.

Choice D

This divides by γ\gamma instead of multiplying, which would make the moving clock appear to run fast rather than slow.

Related formula

Time dilation

Δt=γΔτ,γ=11−v2/c2\Delta t = \gamma\Delta\tau,\quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Assumptions: Both frames are inertial. The proper-time clock is present at both events.

Related topic and practice

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Sources

Authored for this public explainer registry. It has no database question ID and is not copied from a protected or official exam bank.

  1. University Physics Volume 3, Section 5.3: Time Dilation — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
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Sources last checked 2026-08-07

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