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 timeseconds 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
Thus, if it takesseconds to move out to the spot metres from the start point, we have
By the time the second snow plough reaches the same spot, anothermetres of fresh snow would have fallen. Hence
The solution to the differential equation is
Suppose the collision distance is. The time that has elapsed since the snow started falling would be for the second plough and for the first plough.