Question

What is the length contraction of an automobile 3.133 m long when it is traveling at 51.35km/h? (Hint:for x << 1,(1-v2/c2)1/2 ~ 1 - x2/2) Compare this to the diameter of a hydrogen atom by expressing your answer in femto meters.

Answer 1

The length L of a moving object whose rest length is L_0 is:

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

The length contraction can be calculated as the difference L_0-L:

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

For v<, we have:

`\begin{gathered} L_0-L\approx\left(1-\left[1-(1)/(2)\left((v)/(c)\right)^2\right]\right)L_0 \\ \\ =\left((v)/(c)\right)^2(L_0)/(2) \end{gathered}`

Replace v=51.35km/h, c=300,000km/s and L_0=3.133m:

`\begin{gathered} \Delta L=\left((v)/(c)\right)^2(L_0)/(2) \\ \\ =\left((51.35(km)/(h)*(1h)/(3600s))/(300,000(km)/(s))\right)^2(3.133m)/(2) \\ \\ =3.54...*10^(-15)m \\ \\ =3.54...fm \end{gathered}`

The diameter of a hydrogen atom is approximately 10.6*10^-11m. Then, the length contraction of th ecar is much less than the diameter of a hydrogen atom.

Therefore, the length contraction of the automobile is approximately 3.54 femtometers.