A spaceship travels with a speed of 0.3c as it
passes by the Earth on its way to a distant
star, as shown in the diagram below. The
pilot of the spaceship measures the length of
the moving ship as 40 m.
0.3 c
Earth
Determine its length as measured by a person on Earth.
Answer in units of m
β= v/c=0.3c/c=0.3
L=L₀•sqrt{1-β²)=30•sqrt(1-0.3²)=28.6 m
To determine the length of the spaceship as measured by a person on Earth, we can use the concept of length contraction in special relativity. According to special relativity, when an object is moving relative to an observer, its length appears contracted in the direction of motion.
The formula for length contraction is given by:
L' = L * sqrt(1 - v^2/c^2)
Where:
L' is the length as measured by an observer on Earth
L is the length as measured by the pilot on the spaceship
v is the velocity of the spaceship relative to Earth
c is the speed of light
In this case, the velocity of the spaceship relative to Earth is 0.3c, where c is the speed of light. So we can substitute these values into the formula:
L' = 40m * sqrt(1 - (0.3c)^2/c^2)
To simplify the equation, we can replace c with its exact value:
L' = 40m * sqrt(1 - (0.3)^2)
Now, we can calculate the value of L':
L' = 40m * sqrt(1 - 0.09)
L' = 40m * sqrt(0.91)
To find the solution, we can evaluate the square root:
L' = 40m * 0.9539392014169458
L' ≈ 38.16m
Therefore, the length of the spaceship as measured by a person on Earth is approximately 38.16 meters.