Question

as a spaceship is moving toward earth, an earthling measures its length to be 344 m, while the captain on board radios that her spaceship's length is 671 m. calculate the spaceship's speed relative to earth and supply the missing numerical factor below.

Answer 1

The spaceship's speed relative to Earth is approximately 0.625 times the speed of light.

The discrepancy in the measured length of the spaceship between the Earthling observer and the captain on board can be attributed to relativistic effects, specifically length contraction. According to Einstein's theory of special relativity, as an object approaches the speed of light, its length appears contracted when measured by an observer in a different frame of reference.

The Lorentz contraction formula is given by:

`L'=(L)/(r)`

Where:

L ′is the contracted length measured by the Earthling observer,

L is the proper length (length at rest) of the spaceship as communicated by the captain, and

γ is the Lorentz factor, which is dependent on the velocity (v) of the spaceship relative to the Earth.

In this case,

L′=344

L=671 m, and we want to find

v such that:

`344=(671)/(r)`

Solving for γ gives γ≈1.949.

The Lorentz factor is given by:

`r=\frac{1}{\sqrt{1-(v^(2) )/(c^(2) ) } }`

Where

c is the speed of light. Solving for v, we find ≈0.625

v≈0.625c.

Therefore, the spaceship's speed relative to Earth is approximately 0.625 times the speed of light.