Q1; Part A : How fast must a rocket travel on a journey to and from a distant st

Question

Q1; Part A : How fast must a rocket travel on a journey to and from a distant star so that the astronauts age 11.0 while the Mission Control workers on earth age 150 ? Part B: As measured by Mission Control, how far away is the distant star? Q2: A cube has a density of 1800 while at rest in the laboratory. What is the cube's density as measured by an experimenter in the laboratory as the cube moves through the laboratory at 83.0 of the speed of light in a direction perpendicular to one of its faces? Express your answer with the appropriate units

Explanation / Answer

> "Time=ProperTime/sqrt(1-(u/c)^2) but how do I setup the equation to solve for (u/c)^2"

It's pretty much already set up. They give you the "Time" (150 yrs) and the "Proper Time" (10 yrs), so your equation "Time=ProperTime/sqrt(1-(u/c)^2)" now looks like:

150 yrs = (10 yrs) / sqrt(1-(u/c)^2)

Now it's just basic algebra.

Multiply both sides by "sqrt(1-(u/c)^2)":
sqrt(1-(u/c)^2) * (150 yrs) = (10 yrs)

Divide both sides by "150 yrs":
sqrt(1-(u/c)^2) = (10 yrs) / (150 yrs)

Simplify the right side:
sqrt(1-(u/c)^2) = 1/15

Square both sides:
1 - (u/c)^2 = (1/15)^2

Simplify the right side:
1 - (u/c)^2 = 1/225

Add "(u/c)^2" to both sides:
1 = 1/255 + (u/c)^2

Subtract "1/255" from both sides:
1 - 1/255 = (u/c)^2

Simplify the right side:
254/255 = (u/c)^2

Take square root of both sides:
sqrt(254/255) = u/c

So "u/c" is about 0.998

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