1) Let's continue to investigate your space vacation from last week. Your spaceship, which has a proper length of 300 m, passes near a space platform while you are moving at a relative speed of 86% the speed of light (so v=0.86*c). What is the length of the spaceship when measured by someone on the space platform?
Equation for length contraction
L = L' * √ (1 - v2/c2)
Where:
L = The length of the spaceship as seen by the observer
L' = The proper length of the spaceship
v = The velocity of the spaceship
c = The speed of light.
To find the length of the spaceship as measured by someone on the space platform, we can use the equation for length contraction:
L = L' * √(1 - v^2/c^2)
Here:
L' = proper length of the spaceship = 300 m
v = relative speed of the spaceship = 0.86 * c
c = speed of light in a vacuum = 3 x 10^8 meters per second
Now let's substitute the given values into the equation:
L = 300 * √(1 - (0.86c)^2/c^2)
First, let's simplify the expression inside the square root:
(0.86c)^2 = 0.86^2 * c^2 = 0.7396 * c^2
Now we can substitute this back into the equation:
L = 300 * √(1 - 0.7396 * c^2/c^2)
Simplifying further:
L = 300 * √(1 - 0.7396)
L = 300 * √(0.2604)
L = 300 * 0.51016
Finally, we can solve for L:
L ≈ 153.05 meters
Therefore, the length of the spaceship as measured by someone on the space platform is approximately 153.05 meters.