Answer
See answers.
Work Step by Step
$$ \mathcal{l}=\mathcal{l}_0\mathcal{l}_0\sqrt{1-v^2/c^2}$$
Use the binomial expansion to expand the square root expression.
$$\mathcal{l}\approx \mathcal{l}_0(1-\frac{1}{2}v^2/c^2) $$
This was to be proven.
The change in length is the initial length minus the contracted length.
$$\Delta \mathcal{l}=\mathcal{l}_0-\mathcal{l}=\frac{1}{2} \mathcal{l}_0v^2/c^2$$
For the speed given for the train, 100 km/h = 27.78 m/s, so v/c is $9.26\times10^{-8}$.
$$\Delta \mathcal{l}=\frac{1}{2} (500m)( 9.26\times10^{-8})^2=2.14\times10^{-12}m$$
This length shortening is completely unmeasurable, less than the size of a hydrogen atom.
