Starship "Alpha" is traveling at 0.8c (y = 5/3) with respect to the identical starship "Beta." Each starship has rest length 10,000 m. The ship's engineer aboard the "Beta" measures the length of the "Alpha" as it passes by his window. What result should he get?
To determine the length of the starship "Alpha" as measured by the engineer aboard the starship "Beta," we can use the Lorentz transformation equation for length contraction. This equation relates the length of an object in one frame of reference to its length in another frame of reference.
The Lorentz transformation equation for length contraction is:
L' = L * sqrt(1 - v^2 / c^2)
Where:
L' is the contracted length of the object as measured in the "Beta" frame of reference,
L is the rest length of the object,
v is the relative velocity between the two frames of reference,
and c is the speed of light in a vacuum.
In this case, the rest length of both starships "Alpha" and "Beta" is given as 10,000 m. The relative velocity between the two starships is 0.8c, which means v = 0.8 * c.
Plugging these values into the Lorentz transformation equation:
L' = 10,000 * sqrt(1 - (0.8c)^2 / c^2)
= 10,000 * sqrt(1 - 0.64)
= 10,000 * sqrt(0.36)
= 10,000 * 0.6
= 6,000 m
Therefore, the engineer aboard the starship "Beta" should measure the length of the starship "Alpha" as 6,000 meters.