(4 pts.) 1.4 Lorentz Transformation (cf. Probs. 1.6.19+21 in textbook) An observ
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(4 pts.) 1.4 Lorentz Transformation (cf. Probs. 1.6.19+21 in textbook) An observer standing on Earth observes spaceship travel. (a) First, 2 spaceships are heading in the same direction toward Earth, one at v1 = 0.8c and the other one at v2 = 0.85c relative to Earth. What is the speed of spaceship-1 as seen from the captain in spaceship-2? (b) Then 2 spaceships are heading toward Earth from opposite directions, at speeds of U3= 0.9c and v 0.85c relative to Earth. What is the speed of spaceship-4 as seen from the captain in spaceship-3? (c) If spaceship-3 is 0.3ly (1ly-1 lightyear, the distance light travels in one year) away from Earth, how much older (in seconds) will be (i) the observer, and (ii) the captain, once the spaceship-3 lands on Earth? (d) If the captain of spaceship-3 (proper length 450 m) instead decides to pass by Parth at its crusing speed (bs), how long will spaceship-3 be as measured by ceship-3 be as me the observer on Earth?Explanation / Answer
Given
a. speed of spaceship 1 relative to earth, v1 = 0.8c
speed of spaceship wrt earth, v2 = 0.85c
so speed of spaceship one as seen from spaceship 2 is
w = (v1 - v2)/(1 - v1*v2/c^2)
w = (0.8 - 0.85)c/(1 - 0.8*0.85) = -0.15625c
b. when the speeds are in opposite directions
v3 = -0.9c
v4 = 0.85c
v43 = (v4 - v3)/(1 - v4*v3/c^2) = (0.85+0.9)c/(1 + 0.9*0.85) = 0.991c
c. distance of spaceship 3 from earth, d = 0.3 ly
i) time taken by the ship to land on earth as measured by observer, t = d/v3 = 0.3 *c/0.9c = 0.333 years
ii) time takne on the captians frame t' = to*sqroot(1 - v3^2/c^2) = 0.333*sqroot(1 - 0.9^2) = 0.145 years
d. properlength of spaceship 3, lo = 450 m
length measured by an observer on earth, l = lo*sqroot(1 - v3^2/c^2) = 450*sqroot(1 - 0.9^2) = 196.15 m
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