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## #1 2009-04-12 23:31:31

Daniel123
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### 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

## #2 2009-04-13 02:13:25

JaneFairfax
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### 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

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #3 2009-04-13 03:04:35

JaneFairfax
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### 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.

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #4 2009-04-14 07:02:51

Daniel123
Power Member

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### Re: Snow Ploughs

Last edited by Daniel123 (2009-04-14 07:04:21)