LOG#052. Chewbacca’s exam.

I found this fun (Spanish) exam about Special Relativity at a Spanish website:

Solutions:

1) v=25/29 c

2) 1.836 \times 10^{12} m = 12 A.U.

3) t=13.6 months = 13 months and 18 days.

Calculations:

1) We use the relativistic addition of velocities rule. That is,

V=(u-v)/(1-(uv/c^2))

where u=Millenium Falcon velocity, v=imperial cruiser velocity= c/5, y V=relative speed=4c/5.

Using units with c=1:

4/5=(v-1/5)/(1-v/5)

4/5(1-v/5)=v-1/5

4/5-4/25v=v-1/5

29/25 v=1

v=25/29

Then, v=25/29 c reinserting units.

2) This part is solved with the length contraction formula and the velocity calculated in the previous part (1). Moreover, we obtain:

\Delta x'=\Delta x/\gamma

Using the result we got from (1), and plugging that velocity v and the fact that \Delta t' is equal to one hour, then es

\Delta x'=v\Delta t'=\Delta x/\gamma , and from this

\Delta x=\gamma v\Delta t'

Substituting the numerical values, we obtain the given solution easily.

\Delta x =1.97 ( 25/29 c )1hour =1.7 hc=1.836 \times 10^{12} =12A.U.

3) Simple application of time dilation formula provides:

\Delta t'=\gamma \Delta t

Inserting, in this case, our given velocity, we obtain the solution we wrote above:

\Delta t' = 1.97 ( 9 months) = 13.6 months = 13 months 18 days.

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