Spaceship I, which contains students taking a physics exam, approaches Earth with a speed of 0.640 c, while spaceship II, which contains instructors proctoring the exam, moves away from Earth at 0.280 c, as in Figure P26.27. If the instructors in spaceship II stop the exam after 65 min have passed on their clock, how long does the exam last as measured by (a) the students (b) an observer on earth
I've been trying relative velocity addition followed but time dilation but have not gotten the problem right so far
To solve this problem, we will use both the concept of relativistic velocity addition and time dilation.
Let's start by finding the relative velocity between Spaceship I (students on the spaceship) and Earth. We'll denote this relative velocity as v_rel.
Using the formula for relativistic velocity addition, we have:
v_rel = (v_1 + v_2) / (1 + (v_1 * v_2) / c^2)
where:
- v_1 is the velocity of Spaceship I (students) relative to Earth
- v_2 is the velocity of Spaceship II (instructors) relative to Earth
- c is the speed of light in a vacuum (approximately 3.00 x 10^8 m/s)
Given:
- v_1 = 0.640c
- v_2 = -0.280c (negative because Spaceship II is moving away from Earth)
Substituting the given values into the formula, we find:
v_rel = (0.640c - 0.280c) / (1 + (0.640c * -0.280c) / (3.00 x 10^8 m/s)^2)
Calculating, we get:
v_rel = 0.360c / (1 + 0.1792)
≈ 0.360c / 1.1792
≈ 0.305c
Now, let's calculate the time dilation factor, γ, using the formula:
γ = 1 / (sqrt(1 - (v_rel / c)^2))
Substituting the value of v_rel, we have:
γ = 1 / (sqrt(1 - (0.305c / c)^2))
= 1 / (sqrt(1 - 0.305^2))
≈ 1 / (sqrt(1 - 0.093025))
≈ 1 / (sqrt(0.906975))
≈ 1 / 0.952948
≈ 1.049
Now, we can use the time dilation factor to calculate the elapsed time on Spaceship I (students' frame of reference) during the 65 minutes passed on Spaceship II (instructors' frame of reference).
(a) To find the duration of the exam as measured by the students on Spaceship I, we multiply the instructors' time (65 minutes) by the time dilation factor:
Exam duration for the students = 65 min * 1.049
≈ 68.14 min
Therefore, the exam lasts for approximately 68.14 minutes as measured by the students on Spaceship I.
(b) To find the duration of the exam as measured by an observer on Earth, we need to apply time dilation again. This time, we'll use the inverse of the time dilation factor, which represents the time dilation effect from the perspective of Earth:
Exam duration for an observer on Earth = 65 min / 1.049
≈ 61.96 min
Therefore, the exam lasts for approximately 61.96 minutes as measured by an observer on Earth.