# Homework Help: Relative velocity with time/length contraction

1. Apr 8, 2014

If a spaceship travels a distance of one light year as measured from earth in one year's time as measured from the spaceship what is the relative velocity for earth-ship. Also how far did the pilot travel according to the pilot and how long was the trip according to an earth-clock?

I found the distance of a light year. I believe that i have to use gamma as a variable to find u
but i am stuck as i don't know how to get at finding the velocity for the spaceship in order to use the time dilation and length contraction formulas.

any help would be greatly appreciated.

2. Apr 8, 2014

### Staff: Mentor

If the distance is 1 light year as reckoned from earth's frame of reference, what is the distance traveled as reckoned from the spaceship frame of reference (in terms of γ). What is the equation for γ? If the earth traveled this distance (backwards) in 1 year as reckoned from the spaceship frame of reference, what was its speed?

You can also solve this using the Lorentz Transformation. Set Δx = 1 lightyear, Δt' = 1 year, and Δx'=0, and solve for the velocity.

Try it both ways, and verify that you get the same answer.

Chet

3. Apr 9, 2014

i solved for v an got .318724c. can you please confirm this. thanks

4. Apr 9, 2014

### Staff: Mentor

Chet

5. Apr 9, 2014

okay, i used lorentz transformation:
Δx=γ(Δx'+vt) and Δt=γ(Δt'+(vΔx/c^2))
had unknowns of v and Δt so for the first equation i solved for v and got
1ly/√(Δt^2+9.94456e14)
substituted that into the other lorentz equation and got

Δt=√1/(1-(ly/√Δt^2+9.94e14)^2/3e8) * (1+(ly^2/√(Δt^2+9.94e14))/3e16)

solving for Δt yields 3.1535e7 and plugging in that back into my equation for v i got
v=.871343c

6. Apr 9, 2014

now I've used this velocity to find L'
L'=1ly(1-(.873c)^2/(3e^8)^2)^1/2
simplifying i got L'= 5.17681e15m= .547ly as measured by the pilot

now for the time dilation
T=1year(1+(.871c)^2/(3e8)^2= 1.759 years as measured by an earthclock

Last edited: Apr 9, 2014
7. Apr 9, 2014

### dauto

1st: There seem to be something wrong with your Lorentz transform equations. Double check that.
2nd: Do not plug in the value for c until very last step. Plugging in any data from any problem is almost always the very last step of the solution of any problem.

8. Apr 9, 2014

### Staff: Mentor

Yes. I agree with dauto. If you are going to use the Lorentz Transformation, you need to write it down correctly. Your first equation in post #5 is incorrect. When you correct it, that will be the only equation you need to solve this problem.

Chet