a friend speeds by you in her spacecraft at a speed of 0.07c in the positive x direction. the spacecraft is measured in yout frame to be 5m long and 1.5 m high.
1) what will be the spacecraft's length and height at rest?
2)how many seconds would you say elapsed on your friend's watch when 30 sec elapsed on yours?
3)how fast are you travelling, according to your friend?
4)how many seconds would she say elapsed on your watch when she saw 30 sec pass on hers?
A. Lenght :4
Height : 3
To answer these questions, we need to use some concepts from special relativity. Let's go through each question one by one:
1) What will be the spacecraft's length and height at rest?
To find the spacecraft's length and height at rest, we need to consider length contraction due to relativistic effects. The formula for length contraction is given by:
L' = L * √(1 - v²/c²)
where L is the length at rest, L' is the length in the moving frame, v is the velocity of the spacecraft, and c is the speed of light.
In this case, the velocity of the spacecraft is 0.07c (where c is the speed of light), and the length in the frame is 5 meters. Plugging those values into the formula, we get:
L' = 5 * √(1 - (0.07c)²/c²)
= 5 * √(1 - 0.0049)
= 5 * √(0.9951)
≈ 4.99 meters
So, the spacecraft's length at rest will be approximately 4.99 meters. Similarly, we can calculate the height using the same formula.
2) How many seconds would you say elapsed on your friend's watch when 30 sec elapsed on yours?
To calculate the time dilation, we use the formula:
Δt' = Δt / √(1 - v²/c²)
where Δt is the time interval in your frame, Δt' is the time interval in the moving frame, v is the velocity of the spacecraft, and c is the speed of light.
In this case, we have Δt = 30 seconds and v = 0.07c. Plugging those values into the formula, we get:
Δt' = 30 / √(1 - (0.07c)²/c²)
= 30 / √(0.9951)
≈ 30.149 seconds
So, approximately 30.149 seconds would elapse on your friend's watch when 30 seconds elapsed on yours.
3) How fast are you traveling, according to your friend?
To find your velocity according to your friend, we can use the relativistic velocity addition formula:
v' = (v - u) / (1 - v * u / c²)
where v is your velocity, u is the velocity of the spacecraft according to you, and v' is the velocity of you according to your friend.
In this case, u is 0.07c and v' is what we want to find. As we are at rest in our frame, our velocity (v) is 0. Plugging those values into the formula, we get:
v' = (0 - 0.07c) / (1 - 0)
= -0.07c
So, according to your friend, you are traveling with a velocity of approximately -0.07c in the negative x-direction.
4) How many seconds would she say elapsed on your watch when she saw 30 seconds pass on hers?
To calculate the time dilation from your friend's perspective, we can use the same formula as in question 2:
Δt = Δt' / √(1 - v²/c²)
In this case, we know Δt' = 30 seconds and v = -0.07c. Plugging those values into the formula, we get:
Δt = 30 / √(1 - (-0.07c)²/c²)
= 30 / √(1 - 0.0049)
≈ 30.149 seconds
So, approximately 30.149 seconds would elapse on your watch when your friend saw 30 seconds pass on hers.