Both the U and the V problem can be solved using inequality techniques, in slightly different ways. I'll do them below: (spoiler below) (spoiler below) Suppose the downward drop length is a (and the horizontal length is x, and the acceleration by gravity is g). We can work out v at the bottom (sqrt(2ga)) and the time taken to drop (sqrt(2a/g)) using suvat. This gives a time taken of x/sqrt(2g) * (1/sqrt(a)) + 2sqrt(2/g) * sqrt(a) overall. Now, if a = K^4, x/sqrt(2g) = M^2 and 2sqrt(2/g) = N^2, then this quantity is (M^2K^-2 + N^2K^2 = (M/K - NK)^2 + 2MN, which is at least sqrt(x/g) * 2sqrt(2) with equality when a = (M/N)^2 = x/4. Suppose that the angle at the base of the V is 2a, and the horizontal length is x (and the acceleration due to gravity is g). The component of g in the direction of motion is g cos a, and the distance to travel is x/2 cosec a. Now, the time on each branch of the V is given by the suvat equation s = 1/2 at^2 so t^2 = x/g * (1/sin(a)cos(a)) = x/g * (2/sin(2a)). So t(total) = sqrt(8x/g) /sin(2a) >= sqrt(8x/g) with equality iff the angle of the V is a right angle. On Wed, May 16, 2012 at 4:42 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Excellent! What about the corresponding problem where the track is V-shaped? It's a similarly easy exercise with a similarly nice answer -- again, I'd love to know if one can derive it directly without resorting to calculus.
On Wed, May 16, 2012 at 11:22 AM, Veit Elser <ve10@cornell.edu> wrote:
On May 16, 2012, at 10:21 AM, Allan Wechsler wrote:
I was slightly surprised by the answer. I think this would be a fine exercise for elementary differential calculus.
Your problem was on the midterm of my mechanics course this semester.
-Veit
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- James