28.3 Length Contraction

A particle is traveling through the Earth’s atmosphere at a speed of $0 . \text{750} c$. To an Earth-bound observer, the distance it travels is 2.50 km. How far does the particle travel in the particle’s frame of reference?

A spaceship, 200 m long as seen on board, moves by the Earth at $0 . \text{970} c$. What is its length as measured by an Earth-bound observer?

(a) How far does the muon in the reference travel according to the Earth-bound observer? (b) How far does it travel as viewed by an observer moving with it? Base your calculation on its velocity relative to the Earth and the time it lives (proper time). (c) Verify that these two distances are related through length contraction $\gamma= 3 . \text{20}$.

(a) How long does it take the astronaut in the example to travel 4.30 ly at $0 .\text{99944} c$ (as measured by the Earth-bound observer)? (b) How long does it take according to the astronaut? (c) Verify that these two times are related through time dilation with $\gamma= \text{30} . \text{00}$ as given.

(a) Find the value of $\gamma$ for the following situation. An astronaut measures the length of her spaceship to be 25.0 m, while an Earth-bound observer measures it to be 100 m. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?