John Lockard said:
Two futuriestic rockets [R1 and R2] are on a collision course. The rockets are moving
with speeds of 0.800c [v1] and 0.600c [v2] and are initially 2.52E+12 meters [Lo] apart
as measured by Liz, an Earth observer. Both rockets are 50.0 meters [L1' and L2'] in
length as measured by Liz.
Quick calculations
R1: gamma = g1
g1 = 1 / Sqrt(1 - (0.8)^2)
g1 = 1.67
R2: gamma = g2
g2 = 1 / Sqrt(1 - (0.6)^2)
g2 = 1.25
(a) What are their respective proper lengths?
L1' = 50m = L1 / g1 = L1 / 1.67
L1 = 50m * 1.67 = 83.5m
L2' = 50m = L2 / g2 = L2 / 1.25
L2 = 50m * 1.25 = 62.5m
L1 = 83.5m
L2 = 62.5m
(b) What is the length of each rocket as measured by an observer in the other rocket?
As given above:
L1' = 50m
L2' = 50m
(c) According to Liz, how long before the rockets collide?
u = (0.8c + (0.6c))/(1 + (0.8c 0.6c)/c^2)
u = (1.4c) / (1.48)
u = 0.95c = 285000 m/s
To = Lo / u = (2,520,000,000,000 m) / (285,000 m/s)
To = 8,842,105 s
To = 147,368 min
To = 2,456 hours
To = 102 days
(d) According to rocket 1, how long before they collide?
L1 = Lo / g1 = (2,520,000,000,000 m) / (1.67) = 1,509,000,000,000 m
T1 = L1 / u = (1,509,000,000,000 m) / (285,000 m/s)
T1 = 5,295,000 s
T1 = 88,250 min
T1 = 1,470 hours
T1 = 61 days
(e) According to rocket 2, how long before they collide?
L2 = Lo / g2 = (2,520,000,000,000 m) / (1.25) = 2,016,000,000,000 m
T2 = L2 / u = (2,016,000,000,000 m) / (285,000 m/s)
T1 = 7,073,000 s
T1 = 117,883 min
T1 = 1,965 hours
T1 = 82 days
(f) If both rocket crews are capable of total evacuation within 90 minutes (their own time), will there be any casualties?
R1 has time to totally evacuate 980 times.
R2 has time to totally evacuate 1,309 times.
Edited to add: I doubt there will be any casualties. In fact, they probably have enough time to get other ships up there and tow each ship out of collision course or even just to steer out of the way of each other...
(g) What if Franko is right?
If he is, watch out for falling satelites, because NASA has been caclulating their trajectories incorrectly for the last 40 years.
"I love the smell of physics in the morning..."