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 of ms. The speed of each snowplough is ms 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.
There is something exceptionally beautiful about this question, but it is hard to identify exactly what it is; seeing the question for the ﬁrst time makes even hardened mathematicians smile with pleasure.
There is a modelling element to it: you have to set up equations from the information given in the text. The ﬁrst equation you need is a simple ﬁrst order differential equation to ﬁnd the time taken by the ﬁrst snowplough to travel a distance . The corresponding equation for the second snowplough is a bit more complicated, because the depth of snow at any point depends on the time at which the ﬁrst snowplough reached that point, clearing the snow.
The differential equation can be solved using an integrating factor. However, the equation which arises naturally at this point is one involving , which cannot (apparently) be solved by any means. It is the rather good trick of turning the equation upside down (regarding as a function of instead of as a function of ) that allows the problem to be solved so neatly. Apologies if you haven’t come across integrating factors for ﬁrst order differential equations; they are not on our syllabus, but they are really not difficult — you can look online and ﬁnd an easily understandable explanation.
You won’t surprised to learn that there is a generalisation to identical snowploughs .
Solution to problem 52
Suppose that the ﬁrst snowplough reaches a distance at time after it starts snowing. Then the depth of snow it encounters is and its speed is therefore , i.e. . The equation of motion of the ﬁrst snowplough is
Integrating both sides with respect to gives
We know that when (the snowplough started seconds after the snow started), so
This can be rewritten as
When the second snowplough reaches at time , snow has been falling for a time since it was cleared by the ﬁrst snowplough, so the depth at time is metres, i.e. metres. Thus the equation of motion of the second snowplough is
Now we use the standard result (a special case of the chain rule)
to obtain the required equation .
Multiplying by (an integrating factor) and rearranging gives
which integrates to
Since the second snowplough started () at time , the constant of integration is just and the solution is
The snowploughs collide when they reach the same position at the same time. Let this position be . Then
so is given by
This is equivalent to the given formula.