A boy runs a 100m distance in 20s as measured by an observer on earth.how long does it take according to an observer in space ship moving with speed of 0.98c for the boy to complete the run? What is the length of the track according to the moving observer
To solve this problem, we can use the principles of special relativity. According to special relativity, time dilation occurs when an observer is moving relative to an event.
First, let's find the time it takes for the boy to complete the run according to an observer in the spaceship moving with a speed of 0.98c.
The time dilation formula is given by:
Δt' = Δt / √(1 - v^2/c^2)
where Δt' is the time measured by the moving observer, Δt is the time measured by the stationary observer, v is the relative velocity of the two observers, and c is the speed of light.
In this case, Δt is given as 20 seconds and v (relative velocity) is 0.98c.
Plugging these values into the formula:
Δt' = 20 / √(1 - (0.98c)^2/c^2)
Δt' = 20 / √(1 - 0.96^2)
Δt' = 20 / √(1 - 0.9216)
Δt' = 20 / √(0.0784)
Δt' = 20 / 0.280
Δt' ≈ 71.43 seconds
Therefore, according to the observer on the spaceship, it would take approximately 71.43 seconds for the boy to complete the run.
Now let's calculate the length of the track according to the moving observer.
Length contraction occurs when an observer moving relative to an object measures a shorter length.
The formula for length contraction is given by:
L' = L √(1 - v^2/c^2)
where L' is the length measured by the moving observer, L is the length measured by the stationary observer, v is the relative velocity of the two observers, and c is the speed of light.
In this case, L is given as 100 meters and v (relative velocity) is 0.98c.
Plugging these values into the formula:
L' = 100 √(1 - (0.98c)^2/c^2)
L' = 100 √(1 - 0.96^2)
L' = 100 √(1 - 0.9216)
L' = 100 √(0.0784)
L' = 100 √(0.280)
L' ≈ 83.67 meters
Therefore, according to the observer on the spaceship, the length of the track would be approximately 83.67 meters.
To summarize:
- According to the observer in the spaceship moving at a speed of 0.98c, it would take approximately 71.43 seconds for the boy to complete the run.
- According to the moving observer, the length of the track would be approximately 83.67 meters.