The Car and the Train

Let the length of the train=d and the speed of the car x km/hr.

The car travels from rear end to the front and to the rear end again. Total time elapsed=1 hr, since during this time the train has travelled 1 km.
Total time taken by the car=1
So, d/(x-1) + d/(x+1)=1 ( since from rear end to front end relative speed is x-1 and from front to rear end relative speed is x+1).
Solving we get x^2-2dx-1=0

Since the car travelled total 2d distance,then 2d=1km. Substituting this in the original equation we get,
x^2-2x-1=0
Solving we get, x=2+root(4+4)/2 or x=2-root(4+4)/2
Since x>0, x=2+root(4+4)/2
or x=1+root(2).

So the car har travelled x km(total time was 1 hour) i.e. 1+root(2) km.