Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Snow Ploughs

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

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Snow Ploughs

When the first snow plough has been moving for

seconds, the depth of snow at the spot it has reached is

metres so its velocity would be

metres per second. The distance it has moved out would thus be

metres.

Thus, if it takes

seconds to move out to the spot

metres from the start point, we have

By the time the second snow plough reaches the same spot, another

metres of fresh snow would have fallen. Hence

   

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Snow Ploughs

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.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Snow Ploughs

Last edited by Daniel123 (2009-04-13 09:04:21)