Question

How fast (in terms of c) would a 5.9 m long sports car have to be going past you in order for it to appear only 5.4 m long?

Answer 1

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.

Step-by-step 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:

`L=L_0\sqrt{1-(v^2)/(c^2)}`

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:

`\begin{gathered} 5.4\text{ m}=5.9\text{ m}\sqrt{1-(v^2)/(c^2)}\text{ } \\ \\ \frac{5.4\text{ m}}{5.9\text{ m}}=\sqrt{1-(v^2)/(c^2)} \\ \\ (\frac{5.4\text{ m}}{5.9\text{ m}})^2=1-(v^(2))/(c^(2)) \\ \\ (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*0.40 \\ \\ v=0.40c \end{gathered}`

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.