The Relativistic Space Tug · AI Generator #GEN-8C1A1FA2

Physics

Problem

A space salvage company uses a relativistic "tug" spacecraft to rescue disabled vessels. The tug has a powerful ion drive that can maintain constant thrust F=2.40×1012  NF = 2.40 \times 10^{12}\;\text{N} for extended periods. The tug has rest mass m0=8.00×106  kgm_0 = 8.00 \times 10^{6}\;\text{kg} and starts from rest. The tug accelerates under constant thrust until it reaches a speed of 0.800c0.800c, then immediately turns off its engines and coasts at constant velocity to reach a disabled cargo ship. During the coasting phase, mission control on Earth observes the tug for exactly Δt=5.00  years\Delta t = 5.00\;\text{years} (Earth time). (a) What is the proper time Δτ\Delta \tau experienced by the tug's crew during the coasting phase? (b) What distance does the tug travel during the coasting phase, as measured in Earth's reference frame? (c) Due to relativistic effects, the tug's kinetic energy at v=0.800cv = 0.800c is much larger than the classical prediction 12m0v2\frac{1}{2}m_0v^2. Calculate the ratio of the relativistic kinetic energy to the classical prediction.

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