Question

As measured by an observer on the earth, a spacecraft runway on earth has a length of 3600 m. (a) What is the length of the runway as measured by a pilot of a spacecraft flying past at a speed of $\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{*}{\mathbf{10}}^{\mathbf{7}}$m/s relative to the earth? (b) An observer on earth measures the time interval from when the spacecraft is directly over one end of the runway until it is directly over the other end. What result does she get? (c) The pilot of the spacecraft measures the time it takes him to travel from one end of the run- way to the other end. What value does he get?

Short answer

a) The length of the runway as measured by a pilot of a spacecraft is 3568m.

b) The time interval when the spacecraft is directly over one end of the runway until it is directly over the other end is $\mathbf{9}\mathbf{*}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{s}$.

c) The time a pilot takes to travel from one end of the run- way to the other end is $\mathbf{8}\mathbf{.}\mathbf{9}\mathbf{*}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{s}$.

Step-by-step solution

Formulas used to solve the question

Time interval:

$\mathbf{t}\mathbf{=}\frac{\mathbf{∆}{\mathbf{t}}_{\mathbf{0}}}{\sqrt{\mathbf{1}\mathbf{-}{\displaystyle \frac{{\mathbf{u}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}}}$ (1)

Length of object:

$\mathit{l}\mathbf{=}{\mathit{l}}_{\mathbf{0}}\sqrt{\mathbf{1}\mathbf{-}\frac{{\mathbf{u}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}$ (2)

Determine the length of the runway

The proper length is 3600m.

From equation (2), plug the values

$\mathit{l}\mathbf{=}\mathbf{3600}\sqrt{\mathbf{1}\mathbf{-}\frac{{\left(4*{10}^7\right)}^{\mathbf{2}}}{{\left(3*{10}^8\right)}^{\mathbf{2}}}}\mathbf{=}\mathbf{3568}\mathit{m}$

Determine the time interval

It is given by the expression

$\mathbf{∆}\mathit{t}\mathbf{=}\frac{\mathbf{d}}{\mathbf{v}}\mathbf{=}\frac{\mathbf{3600}}{\mathbf{4}\mathbf{*}{\mathbf{10}}^{\mathbf{7}}}\mathbf{=}\mathbf{9}\mathbf{*}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{s}$

Determine the time a pilot takes to travel from one end of the run- way to the other end

From equation (1),

$\mathbf{∆}{\mathit{t}}_{\mathbf{0}}\mathbf{=}\mathbf{∆}\mathit{t}\sqrt{\mathbf{1}\mathbf{-}\frac{{\mathbf{u}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}\mathbf{=}\mathbf{9}\mathbf{*}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\sqrt{\mathbf{1}\mathbf{-}\frac{{\left(4*{10}^7\right)}^{\mathbf{2}}}{{\left(3*{10}^8\right)}^{\mathbf{2}}}}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{9}\mathbf{*}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{s}$

Thus, the length of the runway as measured by a pilot of a spacecraft is 3568m. The time interval when the spacecraft is directly over one end of the runway until it is directly over the other end is $\mathbf{9}\mathbf{*}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{s}$. The time a pilot takes to travel from one end of the run- way to the other end is $\mathbf{8}\mathbf{.}\mathbf{9}\mathbf{*}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{s}$.