Answer:
0.706
Explanation:
Since the other astronaut measures a longer time, this is a time dilation problem. So, our equation for time dilation is given by
T = T₀/√(1 - β²) where T = period on passing space ship = 31.87 s, T₀ = period on other space vehicle = proper time = 22.58 s and β = relative velocity of between the two observers.
T = T₀/√(1 - β²)
√(1 - β²) = T₀/T
squaring both sides, we have
[√(1 - β²)]² = (T₀/T)²
1 - β² = (T₀/T)²
β² = 1 - (T₀/T)²
taking square root of both sides, we have
√β² = √[1 - (T₀/T)²]
β = √[1 - (T₀/T)²]
substituting the values of the variables into the equation, we have
β = √[1 - (22.58 s/31.87 s)²]
β = √[1 - (0.7085)²]
β = √[1 - 0.502]
β = √0.498
β = 0.706
