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 can use the concept of time dilation in special relativity.
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
= 8.3 light-minutes
Next, let's calculate the time dilation factor, γ (gamma), which tells us the time dilation experienced by the moving observer.
γ = 1 / sqrt(1 - (v^2 / c^2))
= 1 / sqrt(1 - (0.6c)^2 / c^2)
= 1 / sqrt(1 - 0.36)
= 1 / sqrt(0.64)
= 1 / 0.8
= 1.25
Now, we can calculate the time difference between the events on the earth and the sun, as observed by the moving observer.
Time difference = γ * (t2 - t1)
= 1.25 * (3 minutes - 0 minutes)
= 1.25 * 3 minutes
= 3.75 minutes
Therefore, the time difference between the events according to an observer moving at u = 0.6c from the earth to the sun is 3.75 minutes.
To calculate the time difference between the events according to an observer moving at 0.6c from the Earth to the Sun, we need to take into account the time dilation effect caused by the observer's motion.
The time dilation formula is given by:
t' = t * sqrt(1 - (v^2 / c^2))
Where:
t' is the time experienced by the moving observer
t is the time measured by a stationary observer
v is the velocity of the moving observer
c is the speed of light in a vacuum
According to the problem, the Earth and the Sun are 8.3 light-minutes apart in the same inertial frame. This means that it takes light 8.3 minutes to travel from the Sun to the Earth.
Now, let's calculate the time difference between the events.
For the observer moving at 0.6c from the Earth to the Sun:
- The time measured by the observer at the Earth (t) is 0 minutes.
- The time measured by the observer at the Sun (t) is 3 minutes.
Using the time dilation formula, we can substitute the values:
t' = 3 * sqrt(1 - (0.6c)^2 / c^2)
t' = 3 * sqrt(1 - 0.6^2)
t' = 3 * sqrt(1 - 0.36)
t' = 3 * sqrt(0.64)
t' = 3 * 0.8
t' = 2.4 minutes
Therefore, according to the observer moving at 0.6c from the Earth to the Sun, the time difference between the events is 2.4 minutes.