Special Relativity - Velocity transformation problem

[seasponges]

Homework Statement

Two spaceships are moving away from Earth at a speed of 0.8c, with one ship following in the flight path of the other. Their separation along the axis of their motion is maintained at 0.1 light years as measures by the spaceships' instruments. A crew exchange vehicle is launched from the rear spaceship to the front spaceship, traveling at 0.9c relative to Earth. Approximately, how long does the trip take for the crew inside the crew exchange vehicle.

Homework Equations

Lorentz Velocity Transformations:

u' = [itex]\frac{u-v}{1-\frac{vu}{c^{2}}}[/itex]

The Attempt at a Solution

u' = [itex]\frac{0.9c-0.8c}{1-0.72}[/itex]
= 0.357c

and

L' = [itex]\sqrt{1-\frac{v^2}{c^2}}[/itex]L[itex]_{0}[/itex]
= 0.093 light years

[itex]\frac{0.093}{0.357}[/itex] = 0.26 years, or 95 days.

However, the question is multiple choice, and the options are:

A. 1 year
B. 25 days
C. 1 day
D. 10 days
E. 150 days

Can someone please point out where I might have slipped up. Thankyou!

 

[Muphrid]

Edit: strike that, I think you did the velocity transformation all right.

What did you plug in for [itex]v[/itex] in the length contraction equation?

 

[seasponges]

0.357c. And v isn't great enough for the length contraction to be the part I messed up on.

 

[Muphrid]

Hm, okay, I worked the problem just in Earth's reference frame and got a different answer, but the calculation is sufficiently involved that I'm not entirely sure on it. Are you familiar with trying to find the point of intersection of two lines? Doing that for the worldlines of the ship and the exchange vehicle may give another useful check.

 

[Nickie]

Your calculations seem correct. However, remember that in special relativity, time is also affected by velocity. This is known as time dilation. The faster an object moves, the slower time passes for that object relative to an observer at rest. In this case, the crew exchange vehicle is traveling at 0.9c relative to Earth, so time will pass slower for the crew inside compared to an observer on Earth. This means that the trip will actually take longer for the crew inside the vehicle.

To calculate the time dilation factor, we use the equation:

\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

Plugging in the velocity of 0.9c, we get \gamma = 2.294. This means that time will pass 2.294 times slower for the crew inside the vehicle compared to an observer on Earth. Therefore, the actual time for the trip would be:

0.26 years x 2.294 = 0.596 years, or approximately 218 days.

This would correspond to option E, 150 days, in the multiple choice question. So your calculations were correct, but you also need to take into account time dilation in this scenario.