Answer :
Given:
The proper length of the sports car is: L₀ = 5.9 m
The contracted length of the sports car is: L = 5.4 m
To find:
The speed of the car.
Explanation:
The length contraction is the shortening of the length of the moving object as measured relative to the observer's frame of reference.
The expression for length contraction is given as:
[tex]L=L_0\sqrt{1-\frac{v^2}{c^2}}[/tex]Here, L is the contracted length, L₀ is the proper length, v is the speed of the object and c is the speed of light having a value 3 × 10⁸ m/s.
Substituting the values in the above equation, we get:
[tex]\begin{gathered} 5.4\text{ m}=5.9\text{ m}\sqrt{1-\frac{v^2}{c^2}}\text{ } \\ \\ \frac{5.4\text{ m}}{5.9\text{ m}}=\sqrt{1-\frac{v^2}{c^2}} \\ \\ (\frac{5.4\text{ m}}{5.9\text{ m}})^2=1-\frac{v^{2}}{c^{2}} \\ \\ \frac{v^{2}}{c^{2}}=1-(\frac{5.4\text{ m}}{5.9\text{ m}})^2 \\ \\ v^2=c^2(1-(\frac{5.4\text{ m}}{5.9\text{ m}})^2) \\ \\ v=\sqrt{c^2(1-(\frac{5.4\text{ m}}{5.9\text{ m}})^2)} \\ \\ v=c\sqrt{1-\frac{(5.4\text{ m\rparen}^2}{(5.9\text{ m\rparen}^2}} \\ \\ v=c\times0.40 \\ \\ v=0.40c \end{gathered}[/tex]Final answer:
The sports car should travel with a speed of 0.40c so that the length of the car appears to be 5.4 m.