Time Dilation, Length Contraction and Energy

Proper time t₀ is the interval between two events measured in a frame where they occur at the same place, such as a particle’s creation and decay in its rest frame. An observer who sees the clock moving measures a longer interval. Proper length l₀ is measured in the object’s rest frame; a frame seeing it move measures a shorter length along its motion.

The two ends of a moving object must be located simultaneously in the measuring frame. Contraction applies along the direction of relative motion, not equally in every direction. Define which frame measures each quantity before substituting numbers.

Fast muons created high in the atmosphere reach the ground in much greater numbers than a non-relativistic lifetime estimate predicts. In Earth’s frame their lifetime is dilated, allowing a larger travel distance.

In the muon’s frame its proper lifetime is unchanged, but the distance through the atmosphere is contracted. These are compatible descriptions of the same events, not two independent corrections to multiply into one frame’s calculation.

AQA sometimes expresses relativistic mass as m=γm₀. Equivalently retain invariant rest mass m₀ and write total energy $E=\gamma m_0c^2$. Rest energy is $m_0c^2$ and kinetic energy is $(\gamma-1)m_0c^2$. As speed approaches c, γ and kinetic energy rise without bound; a massive particle does not reach c with finite energy.

Bertozzi accelerated electrons and compared energy input with their measured time of flight. Increasing energy eventually produced only a small further speed increase, in disagreement with the classical prediction $E_k=mv^2/2$ at those energies and consistent with relativistic energy.

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.