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:

$L=\sqrt{1-\frac{v^2}{c^2}}L_0$

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

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

For v<, we have:

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

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

$\begin{gathered} \Delta L=\left(\frac{v}{c}\right)^2\frac{L_0}{2} \\ \\ =\left(\frac{51.35\frac{km}{h}\times\frac{1h}{3600s}}{300,000\frac{km}{s}}\right)^2\frac{3.133m}{2} \\ \\ =3.54...\times10^{-15}m \\ \\ =3.54...fm \end{gathered}$

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.