From: gelfand@lamar.ColoState.EDU (Martin Gelfand)
Newsgroups: sci.math
Subject: Re: Help: Snowplow problem
Date: 27 Sep 1995 12:14:02 -0600
In article Kevin Shack writes:
>This problem is supposed to be solvable, given a couple assumptions:
>
>It starts snowing at a heavy but constant rate. A snow plow starts at
>noon. In the first hour, the plow travels 2 miles. In the second hour,
>the plow travels 1 mile. What time did it start snowing?
>
>Any solutions?
>
Sure, I have a solution: to the nearest ten minutes, it started
at twenty after eleven in the morning. I'm not going to
be more precise, and I have no intention of solving anyone's
homework problems for them.
However, here are some hints towards the solution, which are
related to the "couple assumptions" one needs to make.
You have to make some assumption about how quickly the snow
is falling, as a function of time. The simplest assumption
is that it falls at a constant rate (once it starts falling).
You have to make some assumption about how the speed of
the plow depends on the depth of the snow. If the plow's
speed was independent of snow depth then it would travel
as far in the first hour as the second hour, so that's
not a good assumption to make. In the classic snowplow problem
one assumes that the plow's speed varies as 1/(snow depth).
This has a very unphysical feature: the plow can move infinite
fast over a clear road! However, as long as the snow depth
is finite the plow's motion is too not unreasonable; and it's
this particular form for the plow's speed that permits an
analytical solution of the problem.
So, construct the appropriate differential equation, and go forth!
Martin Gelfand
Dept of Physics,
Colorado State University
==============================================================================
From: drourke@news.sfu.ca (David Rourke)
Newsgroups: sci.math
Subject: Re: Help: Snowplow problem
Date: 28 Sep 1995 04:39:18 GMT
gelfand@lamar.ColoState.EDU (Martin Gelfand) writes:
>You have to make some assumption about how the speed of
>the plow depends on the depth of the snow. If the plow's
>speed was independent of snow depth then it would travel
>as far in the first hour as the second hour, so that's
>not a good assumption to make. In the classic snowplow problem
>one assumes that the plow's speed varies as 1/(snow depth).
>This has a very unphysical feature: the plow can move infinite
>fast over a clear road! However, as long as the snow depth
>is finite the plow's motion is too not unreasonable; and it's
>this particular form for the plow's speed that permits an
>analytical solution of the problem.
Actually, I believe the correct assumption to make is that the
snowplow can remove a certain constant amount of snow per time unit.
As the snow depth increases, the snowplow must therefore slow down.
Dave.
==============================================================================
Newsgroups: sci.math
From: Dave Dodson
Subject: Re: When did it start to snow? SPOILER
Date: Wed, 21 Oct 1992 20:46:23 GMT
In article pmanne@math.uio.no (Per Manne) writes:
>Here is a nasty calculus exercise which was given in an exam many, many
>years ago with disastrous consequences.
>
>Three snowplows share responsibility for the same road. The speed of a
>snowplow is inversely proportional to the amount of snow in front of it.
>Sometime before noon it starts snowing. At noon the first snowplow starts
>plowing the road. It continues to snow (with the same intensity), and one
>hour later the second snowplow starts plowing. At two o'clock the last
>snowplow starts plowing (all snowplows start at the same point and go in
>the same direction). After some time (an hour or so) the third snowplow
>catches up with the second, and at the same moment the second snowplow
>catches up with the first.
>
>When did it start to snow?
Instead of finding distance s as a function of time t, I will find t as
a function of s. Assume it starts snowing t_0 hours before noon, and snows
at a rate of 1 unit per hour. Then at time t_1, the first snowplow faces
snow of depth t_0 + t_1. We assume that the plow can move at 1 unit/hour
in snow 1 foot deep, making the constant of proportionality equal to 1, so
the speed of the plow obeys (dt_1/ds) = t_0 + t_1 with intial condition
t_1 = 0 at s = 0. The solution to this differential equation is
t_1 = t_0 * exp(s) - t_0
At time t_2, the second snowplow faces snow t_2(s) - t_1(s) deep, so it
obeys (dt_2/ds) = t_2 - t_0 * exp(s) _ t_0 with initial condition t_2 = 1
at s = 0. The solution to this differential equation is
t_2 = (1 + t_0 - t_0 * s) * exp(s) - t_0
At time t_3, the third snowplow faces snot t_3(s) - t_2(s) deep, so it
obeys (dt_3/ds) = t_3 - (1 + t_0 - t_0 * s) * exp(s) + t_0 with initial
condition t_3 = 2 at s = 0. The solution to this differential equation is
t_3 = (2+t_0 - (1+t_0)*s + t_0/2*s^2) * exp(s) - t_0
Now letting the first and second plows colide at s = S:
t_0 * exp(S) - t_0 = (1 + t_0 - t_0 * S) * exp(S) - t_0
or
t_0 * S = 1
If the third plow is at the same point at the same time then
t_0 * exp(S) - t_0 = (2+t_0 - (1+t_0)*S + t_0/2*S^2) * exp(S) - t_0
or
0 = 2 - (1 + t_0) * S + t_0/2 * S^2
Inserting S = 1/t_0 from above, we get
0 = 2 - (1 + t_0)/t_0 + 1/(2 * t_0)
from which
t_0 = 1/2
Hence, it started snowing at 11:30.
----------------------------------------------------------------------
Dave Dodson dodson@convex.COM
Convex Computer Corporation Richardson, Texas (214) 497-4234