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The problem:

One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going 2 miles the first hour and 1 mile the second hour. What time did it start snowing?

I know the answer to this problem and how it is usually approached. But I was wondering if there was another way to solve it.

If c1 equals the change in acceleration due to increasing resistance from the snow then I get the following four equations:

d3x/dt3 = c1

d2x/dt2 = c1t + c2

dx/dt = (1/2)c1t2 + c2t + c3

x = (1/6)c1t3 + (1/2)c2t2 + c3t + c4

Where (0,0),(1,2),(2,3) .

The problem is I don't have enough points to solve for c1 and c2, which I think are the only two constants I need so that I can set acceleration equal to zero and solve that equation. Is there something I am missing here, or am I way off the mark?

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Your model does not reflect the idea that the plow removes equal amounts of snow in equal times, but stipulates that its velocity decreases linearly with time. This would imply that the plow comes to halt at a certain moment, and then starts moving backwards. –  Christian Blatter Aug 9 at 18:32

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