At what time did the snow start to fall?
Question by Track P: At what time did the snow start to fall?
On a certain winter day, snow starts to fall at a heavy and steady rate. Three identical snowplows start plowing the same road, the first leaving at 12 noon, the second leaving at 1 pm, and the third leaving at 2 pm. At some time later, they all collide. At what time did the snow start to fall?
The speed of a snowplow is inversely proportional to the depth of the snow.
Best answer:
Answer by illuminatiposer
when u start using dove
Add your own answer in the comments!
Are we assuming that all 3 plows are following the same exact road, and thus plowing over what has been accumulated since the last truck came by? Or we assuming that each truck is plowing over completely unplowed snow?
You are leaving something out…
How can the speed be inversely proportional without telling us, for example, what the starting speed of the truck is, etc.
Example; if the truck starts out going 20 mph, and one inch of snow falls, what does that mean as far as speed? Does the speed reduce by 50%..we don’t know what the expected total snowfall will be or the starting speed of the truck, so you can’t figure the speed that way…(inverse to what?)
I’m guessing you saw/read this somewhere but you are not providing all the info to answer…
Well, I found where the question is from…and it appears you did COPY the info provided, but i’m still confused…”uncle”!!!
http://www.phy.duke.edu/~hsg/physics-challenges/challenges.html
This is a good example of how math does not solve all…double all factors in the formula that “V” entered, and it changes the “time” doesn’t it?? The question is incomplete in order to answer…you can (as has been done) provide a formula to answer given one or more variables, but given the info provided it cannot be answered accurately.
Yes Vasek, I was referring to you, but your name disappeared when I tried to post. I did not mean any disrespect.
You very cleary are quite a knowledgeable fellow, and I don’t dispute your formula, however, I guess by “three identical snowplows” I did not assume that they were necessarily traveling the same speed. That’s where my concern came in..perhaps I read to much into the question, but I would have phrased the question (rather than say three identical snowplows) “three snowplows traveling the exact same speed”…
Again, no disrespect to you or the question asker, I just read the question differently.
At 11:30 AM.
To solve this, I made three unknown functions marking time the three snowplows pass a certain position s: t1(s), t2(s), t3(s). For uniformity, let’s also define t0(s) to be a constant function of the time when the snow began to fall.
The i-th snowplow passing at the point s “sees” all the snow that fell there between t_(i-1) and t_i. According to your condition,
v = ds / dt_i = c/(t_i – t_(i-1)) for some constant c
=>
dt_i / ds = (t_i – t_(i-1))/c
For simplicity, let’s denote b = 1/c.
The initial conditions are:
t0(s) = time when the snow began to fall
t1(0) = 12 AM
t2(0) = 1 PM
t3(0) = 2 PM
We can shift the time coordinate so that
t0(s) = 0 for all s
t1(0) = a
t2(0) = a + 1
t3(0) = a + 2
Now a means for how long the snow had been falling at the noon.
Let’s solve this system.
t1′ = b*(t1 – t0) = b*t1
t1(s) = C1*exp(bs)
t1(0) = C1 = a =>
t1(s) = a*exp(bs)
t2′ = b*(t2-t1) = b t2 – ab exp(bs)
variation of constants: t2(s) = f(s)*exp(bs), f’(s) = -ab
t2(s) = (C2 – abs) exp(bs)
t2(0) = C2 = a + 1 =>
t2(s) = (a + 1 – abs) exp(bs)
t3′ = b*(t3-t2) = b t3 – (ab + b – ab^2 s) exp(bs)
t3(s) = g(s)*exp(bs), g’(s) = -(ab + b – ab^2 s)
t3(s) = (C3 – abs – bs + ab^2s^2/2) exp(bs)
t3(0) = C3 = a + 2 =>
t3(s) = (a + 2 – abs – bs + ab^2s^2/2) exp(bs)
Now, all of t1(s), t2(s) and t3(s) should be the same at some point s. This gives a system of equations
a = a + 1 – abs = a + 2 – abs – bs + ab^2s^2/2
with unknowns a and s. We are not actually interested in s, a is the main question and all other letters are some constants defined above.
The first = sign gives abs = 1, plugging this into the second one and eliminating s yields
a = a + 2 – 1 – 1/a + 1/(2a)
0 = 1 – 1/(2a)
2a = 1
independently of b.
Therefore, a = 1/2 hour, so the snow started to fall at 11:30 AM.
Nice question!
@ jason s: If “V” means my name, please note that the rate of the snowfall, the constant b, was effectively lost during the manipulations and the answer does NOT depend on it.