A lovely problem, taken from Advanced Problems in Mathematics, by Dr Siklos*

Two identical snowploughs plough the same stretch of road. The first starts at a time

seconds after it starts snowing, and the second starts from the same point

seconds later, going in the same direction. Snow falls so that the depth of snow increases at a constant rate

. The speed of each snowplough is

, where

is the depth (in metres) of the snow it is ploughing, and

is a constant. Each snowplough clears all the snow. Show that the time

at which the second snowplough has travelled a distance

metres satisfies the equation:

. Hence show that the snowploughs will collide when they have travelled

metres.

*He adds in his discussion that this can be generalised to n identical snowploughs