a spaceship travels with a speed of 0.7c as it passes by the Earth on its way to a distant star. The pilot of the spaceship measures the length of the moving ship as 50m. Determine its length as measured by a person on Earth. Answer in units of m.
for this question, do I just find the gamma using 0.7c and then 50*gamma to get the length?
yes.
To find the length of the spaceship as measured by a person on Earth, you can use the time dilation formula:
γ (gamma) = 1 / sqrt(1 - (v^2 / c^2))
Where:
γ (gamma) is the Lorentz factor,
v is the velocity of the spaceship (0.7c),
and c is the speed of light (299,792,458 m/s).
First, calculate γ using the given velocity:
γ = 1 / sqrt(1 - (0.7c)^2 / c^2)
γ = 1 / sqrt(1 - 0.49)
γ = 1 / sqrt(0.51)
γ ≈ 1.22474
Next, multiply the length measured by the pilot (50m) by γ to find the length as measured by a person on Earth:
Length on Earth = 50m * γ
Length on Earth ≈ 50m * 1.22474
Length on Earth ≈ 61.237 m
Therefore, the length of the spaceship as measured by a person on Earth is approximately 61.237 meters.