Respuesta :
Answer:
a. speed, v = 0.97 c
b. time, t' = 20.56 years
Given:
t' = 5 years
distance of the planet from the earth, d = 10 light years = 10 c
Solution:
(a) Distance travelled in a round trip, d' = 2d = 20 c = L'
Now, using Length contraction formula of relativity theory:
[tex]L'' = L'\sqrt{1 - \frac{v^{2}}{c^{2}}}[/tex] (1)
time taken = 5 years
We know that :
time = [tex]\frac{distance}{speed}[/tex]
5 = [tex]\frac{L''}{v}[/tex] (2)
Dividing eqn (1) by v on both the sides and substituting eqn (2) in eqn (1):
[tex]\frac{L'\sqrt{1 - \frac{v^{2}}{c^{2}}}}{v} = 5[/tex]
[tex]\frac{20'\sqrt{1 - \frac{v^{2}}{c^{2}}}}{v} = 5[/tex]
Squaring both the sides and Solving above eqution, we get:
v = 0.97 c
(b) Time observed from Earth:
Using time dilation:
[tex]t'' = \frac{t'}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}[/tex]
[tex]t'' = \frac{5}{\sqrt{1 - \frac{(0.97c)^{2}}{c^{2}}}}[/tex]
Solving the above eqn:
t'' = 20.56 years
