Answer:
The value is [tex] v =1.58 *10^{8} \ m/s[/tex]
Explanation:
From the question we are told that
The length of the spacecraft streaks is [tex]L_i = 85 \ m[/tex]
The proper length of the spacecraft streaks [tex]L_p = 100 \ m[/tex]
Generally from the equation of relativity of length contraction
[tex]L_i = L_p * \sqrt{1 - \frac{v^2}{c^2} }[/tex]
Here c is the speed of light with value [tex]c = 3.0 *10^{8} \ m/s^2[/tex]
v is the velocity of the spacecraft
So
[tex]\frac{v^2}{c^2} = 1 - \frac{L_i^2}{L_p^2 }[/tex]
=> [tex]\frac{v^2}{c^2} = \frac{L_p^2 - L_i^2}{L_p^2}[/tex]
=> [tex]\frac{v^2}{c^2} = \frac{100^2 - 85^2}{100^2}[/tex]
=> [tex]\frac{v^2}{c^2} = 0.2775[/tex]
=> [tex] v = \sqrt{0.2775c^2}[/tex]
=> [tex] v = 0.5268c[/tex]
=> [tex] v = 0.5268 * 3.0*10^{8}[/tex]
=> [tex] v =1.58 *10^{8} \ m/s[/tex]
