gen-define-time-dilation-and-give-one-short-worked-example-ec35

Define time dilation and give one short worked example.

# Time Dilation **Time dilation** is the phenomenon where time passes at different rates for observers in different reference frames. Specifically, a moving clock runs slower than a stationary one, as measured by the stationary observer. The relationship is given by: $$\Delta t = \frac{\Delta t_0}{\sqrt{1-v^2/c^2}} = \gamma \Delta t_0$$ where: - Δt₀ = proper time (time in the moving object's frame) - Δt = dilated time (time measured by stationary observer) - v = relative velocity - c = speed of light - γ = Lorentz factor ## Worked Example **Problem:** A spacecraft travels at 0.6c relative to Earth. If 10 years pass on the spacecraft, how much time passes on Earth? **Solution:** First, find the Lorentz factor: $$\gamma = \frac{1}{\sqrt{1-v^2/c^2}} = \frac{1}{\sqrt{1-(0.6)^2}} = \frac{1}{\sqrt{0.64}} = \frac{1}{0.8} = 1.25$$ Then calculate Earth time: $$\Delta t = \gamma \Delta t_0 = 1.25 \times 10 \text{ years} = 12.5 \text{ years}$$ **Answer:** 12.5 years pass on Earth while only 10 years pass on the spacecraft.