[tex]3.20×10^7\:\text{m/s}[/tex]
Explanation:
Let
[tex]L = 28.2\:\text{m}[/tex]
[tex]L' = 28.2\:\text{m} - 0.161\:\text{m} = 28.039\:\text{m}[/tex]
The Lorentz length contraction formula is given by
[tex]L' = L\sqrt {1 - \left(\dfrac{v^2}{c^2}\right)}[/tex]
where L is the length measured by the moving observer and L' is the length measured by the stationary Earth-based observer. We can rewrite the above equation as
[tex]\sqrt {1 - \left(\dfrac{v^2}{c^2}\right)} = \dfrac{L'}{L}[/tex]
Taking the square of the equation, we get
[tex]1 - \left(\dfrac{v^2}{c^2}\right) = \left(\dfrac{L'}{L}\right)^2[/tex]
or
[tex]1 - \left(\dfrac{L'}{L}\right)^2 = \left(\dfrac{v}{c}\right)^2[/tex]
Solving for v, we get
[tex]v = c\sqrt{1 - \left(\dfrac{L'}{L}\right)^2}[/tex]
[tex]\:\:\:\:=(3×10^8\:\text{m/s})\sqrt{1 - \left(\dfrac{28.039\:\text{m}}{28.2\:\text{m}}\right)^2}[/tex]
[tex]\:\:\:\:=3.20×10^7\:\text{m/s} = 0.107c[/tex]
