The earth and the sun are 8.3 light-minutes apart in the same inertial frame. Two events occur at t = 0 on the earth and t = 3 minutes on the sun respectively. Calculate the time difference between the events according to an observer moving at u = 0.6c from the earth to the sun. (1 light minute is the distance travelled by light in one minute.)
To calculate the time difference between the events according to an observer moving at u = 0.6c from the earth to the sun, we need to take into account the time dilation effect due to relativistic motion.
First, let's calculate the time it takes for light to travel from the earth to the sun, which is given as 8.3 light-minutes.
Time taken by light = distance / speed of light
Time taken by light = 8.3 light-minutes / 1 light-minute per minute = 8.3 minutes
So, it takes 8.3 minutes for light to travel from the earth to the sun.
Now, let's calculate the time difference between the events according to the observer moving at u = 0.6c.
According to time dilation formula, Δt' = Δt / γ
Where:
Δt' is the proper time interval observed in the moving frame (observer moving at u = 0.6c)
Δt is the time interval observed in the stationary frame (earth frame, where events occur at t = 0 and t = 3 minutes)
γ is the Lorentz factor, given by γ = 1 / sqrt(1 - (v/c)^2), where v is the relative velocity between the frames and c is the speed of light
Given:
Δt = 3 minutes
Now, let's calculate γ:
γ = 1 / sqrt(1 - (u/c)^2)
γ = 1 / sqrt(1 - (0.6c/c)^2)
γ = 1 / sqrt(1 - 0.36)
γ = 1 / sqrt(0.64)
γ = 1 / 0.8
γ = 1.25
Now, let's calculate Δt':
Δt' = Δt / γ
Δt' = 3 minutes / 1.25
Δt' = 2.4 minutes
Therefore, according to an observer moving at u = 0.6c from the earth to the sun, the time difference between the events is approximately 2.4 minutes.