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, you are on the right track! To determine the length of the spaceship as measured by a person on Earth, you can use the concept of length contraction.
Length contraction refers to the phenomenon that an object's length appears shorter when it is moving relative to an observer. The formula for length contraction is:
L' = L / γ
Where:
L' is the contracted length as measured by the observer (in this case, a person on Earth),
L is the length measured by the moving observer (in this case, the pilot of the spaceship),
γ (gamma) is the Lorentz factor, which is given by:
γ = 1 / √(1 - (v²/c²))
In this case, the spaceship travels at a speed of 0.7c, so its velocity is 0.7 times the speed of light. Therefore, we can plug in these values into the formulas and calculate the contracted length:
First, calculate the Lorentz factor:
γ = 1 / √(1 - (0.7²/1²))
= 1 / √(1 - 0.49)
= 1 / √(0.51)
≈ 1 / 0.7141
≈ 1.400
Now, plug in the calculated Lorentz factor into the length contraction formula:
L' = 50 / γ
≈ 50 / 1.400
≈ 35.71
Therefore, the length of the spaceship as measured by a person on Earth would be approximately 35.71 meters.