Question

From a laboratory on Earth, you observe a spacecraft traveling horizontally at a speed of 0.580 𝑐. You measure it to have a horizontal length of 5.80 𝑚 and a vertical height of 1.20 𝑚. What will be its length and height if it comes to rest in the lab?

 Length (m)  Height (m)

(A) 7.12 1.47
(B) 7.12 1.20
(C) 4.75 1.20
(D) 4.75 0.978

Answer 1

`(B)`

Explanation:

First let's work out the gamma factor `\gamma` which is given by the equation

`gamma = (1)/(sqrt(1-v^2/c^2))`

`gamma = (1)/(sqrt(1-0.58^2))`

`gamma = 1.22757`

Then, we can use the formula for length contraction which is...

`L = L_0 / gamma`

where `L_0` is the reference frame where the spacecraft is at rest in front of the observer in the lab hence...

`L_0 = gammaL`

Because the observer sees the spacecraft moving at speeds relative to the speed of light, in his reference frame, he says a contracted length. Which means that we must get a length that is larger than the observed one when the spacecraft is at rest in the lab. We can use this with the lengths we already have to get...

`L_0 = 1.23 xx 5.8`

`=> L_0 = 7.13\text(m)`

Furthermore, because the aircraft is travelling, the observer from Earth observers the proper vertical length of the aircraft. Length contraction only happens along the direction of travel. Therefore, the vertical height remains `1.20\text(m)`.

This means the answer is `(B)`