28.3 Length contraction

A particle is traveling through the Earth’s atmosphere at a speed of $0\text{.}\text{750}c$ . 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?

To whom does an object seem greater in length, an observer moving with the object or an observer moving relative to the object? Which observer measures the object’s proper length?

Relativistic effects such as time dilation and length contraction are present for cars and airplanes. Why do these effects seem strange to us?

Suppose an astronaut is moving relative to the Earth at a significant fraction of the speed of light. (a) Does he observe the rate of his clocks to have slowed? (b) What change in the rate of Earth-bound clocks does he see? (c) Does his ship seem to him to shorten? (d) What about the distance between stars that lie on lines parallel to his motion? (e) Do he and an Earth-bound observer agree on his velocity relative to the Earth?

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

How fast would a 6.0 m-long sports car have to be going past you in order for it to appear only 5.5 m long?

(a) How far does the muon in [link] 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 $\text{γ=}3\text{.}\text{20}$ .

(a) How long would the muon in [link] have lived as observed on the Earth if its velocity was $0\text{.}\text{0500}c$ ? (b) How far would it have traveled as observed on the Earth? (c) What distance is this in the muon’s frame?

(a) How long does it take the astronaut in [link] 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 $\text{γ=}\text{30}\text{.}\text{00}$ as given.

(a) How fast would an athlete need to be running for a 100-m race to look 100 yd long? (b) Is the answer consistent with the fact that relativistic effects are difficult to observe in ordinary circumstances? Explain.

Unreasonable Results

(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?

Unreasonable Results

A spaceship is heading directly toward the Earth at a velocity of $0\text{.}\text{800}c$ . The astronaut on board claims that he can send a canister toward the Earth at $1\text{.}\text{20}c$ relative to the Earth. (a) Calculate the velocity the canister must have relative to the spaceship. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?