Muons are unstable particles with a proper lifetime of 2.2 μs. Muons are produced 2.0 km above ground and move downwards at a speed of 0.98c relative to the ground. For this speed \(\gamma \) = 5.0. Discuss, with suitable calculations, how this experiment provides evidence for time dilation.

ALTERNATIVE 1 — for answers in terms of time
overall idea that more muons are detected at the ground than expected «without time dilation»

«Earth frame transit time = \(\frac{{2000}}{{0.98c}}\)» = 6.8 «μs»

«Earth frame dilation of proper half-life = 2.2 μs x 5» = 11 «μs»
OR
«muon’s proper transit time = \(\frac{{6.8\mu s}}{5}\)» = 1.4 «μs»

ALTERNATIVE 2 – for answers in terms of distance
overall idea that more muons are detected at the ground than expected «without time dilation»

«distance muons can travel in a proper lifetime = 2.2 μs x 0.98c» = 650 «m»

«Earth frame lifetime distance due to time dilation = 650 m x 5» = 3250 «m»
OR
«muon frame distance travelled = \(\frac{{2000}}{5}\)» = 400 «m»

Accept answers from one of the alternatives.

[3 marks]