Astrotrain, a Decepicon triple changer, is traveling at a speed of .77c towards Earth, as seen by Blaster, who is on Earth inside Fortress Maximus. Astrotrain's computers tell him the icon rocket was fired at .35c. Determine the length of the ion rocket as seen by Blaster and Fortress Maximus.
Um, how can one determine a relativistically affected length, if no length at all is given in the problem?
I apoligize. I forgot to include that Astrotrain fired a 3m ion rocket at Fotress Maximus.
The speed of the rocket relative to Earth must first be computed. There is a relavistic velocity-addition formula for that. You can find it at
http://en.wikipedia.org/wiki/Velocity-addition_formula
The value for the relative velocity is
v = (.77c + 0.35c)/[1 + (0.77*0.35)]
= 0.882 c
Now use the Fitz-Gerald contraction formula for the apparent length of the rocket, L', as seen from Earth
L' = L * sqrt[1 - (v/c)^2]= 0.471 L
= 1.41 m
To determine the length of the ion rocket as seen by Blaster and Fortress Maximus, we can use the relativistic velocity addition formula. According to this formula, the relative velocity between two objects moving in the same direction is given by:
V = (v1 + v2) / (1 + (v1 * v2) / c^2)
Where:
V is the relative velocity between the two objects.
v1 is the velocity of Astrotrain.
v2 is the velocity of the ion rocket.
c is the speed of light in a vacuum.
In this case, Astrotrain's velocity is 0.77c, and the ion rocket's velocity is 0.35c. Let's substitute these values into the formula:
V = (0.77c + 0.35c) / (1 + (0.77c * 0.35c) / c^2)
V = (1.12c) / (1 + (0.77 * 0.35))
V = (1.12c) / (1 + 0.2695)
V = 1.12c / 1.2695
V ≈ 0.88297c
Now, to calculate the length contraction, we use the Lorentz transformation formula:
L' = L / γ
Where:
L' is the length of the ion rocket as seen by Blaster and Fortress Maximus.
L is the rest length of the ion rocket.
γ is the Lorentz factor, given by γ = 1 / sqrt(1 - v^2 / c^2)
In this formula, v is the relative velocity of the ion rocket with respect to Blaster and Fortress Maximus. Let's substitute the values into the formula:
v = 0.88297c
γ = 1 / sqrt(1 - (0.88297c)^2 / c^2)
γ = 1 / sqrt(1 - 0.7805)
γ = 1 / sqrt(0.2195)
γ = 1 / 0.4683
γ ≈ 2.136
Finally, we can calculate the length contraction:
L' = L / γ
Since we don't have the rest length of the ion rocket (L), we can't determine the exact length as seen by Blaster and Fortress Maximus without additional information.