A rocket blasts off from earth to a near star travelling at 0.80c. If the star is 20 light years away, how much time elapses on earth ? In the rocket ?
To calculate the time elapsed on Earth and in the rocket, we can use the concept of time dilation from special relativity. According to the theory, moving clocks will appear to be running slower relative to stationary clocks.
First, let's calculate the time elapsed on Earth. The distance to the star is given as 20 light years. Since the rocket is traveling at 0.80c, which means it is moving at 80% of the speed of light, we need to take into account time dilation.
The time dilation factor, γ (gamma), is given by the equation:
γ = 1 / √(1 - v²/c²)
Where:
v is the velocity of the rocket (0.80c in this case)
c is the speed of light (approximately 3 x 10^8 meters per second)
Substituting the values into the equation, we get:
γ = 1 / √(1 - 0.80²)
Simplifying, we find:
γ = 1 / √(1 - 0.64)
≈ 1 / √(0.36)
≈ 1 / 0.6
≈ 1.67
This means that time on the rocket will be dilated by a factor of approximately 1.67 compared to Earth.
Now, let's calculate the time elapsed on Earth.
The time taken to travel the distance to the star, as observed from Earth, can be calculated by dividing the distance by the velocity of light:
Time Elapsed on Earth = Distance / Velocity
= 20 light years / (3 x 10^8 m/s)
≈ 20 x (9.46 x 10^15 m) / (3 x 10^8 m/s)
≈ 6.32 x 10^16 seconds
Since the time dilation factor is 1.67, the time elapsed on the rocket can be calculated by multiplying the time elapsed on Earth by the time dilation factor:
Time Elapsed on Rocket = Time Elapsed on Earth * Time Dilation Factor
≈ (6.32 x 10^16 seconds) * 1.67
≈ 1.06 x 10^17 seconds
So, approximately 6.32 x 10^16 seconds (on Earth) and 1.06 x 10^17 seconds (on the rocket) will elapse during the journey to the star.