To a close approximation, the pressure behind a rifle bulletvaries inversely wit

Question

To a close approximation, the pressure behind a rifle bulletvaries inversely with the position x of the bullet along thebarrel. Thus the acceleration of the bullet may be written asa=k/x, where k is a constant. If the bullet starts at rest atx=7.5mm and if the muzzle velocity of the bullet is 600m/s at theend of the the 750mm barrel, compute the acceleration of the bulletas it passes the midpoint of the barrel at x=375mm. - The answer given is :104km/s2. Please Help! To a close approximation, the pressure behind a rifle bulletvaries inversely with the position x of the bullet along thebarrel. Thus the acceleration of the bullet may be written asa=k/x, where k is a constant. If the bullet starts at rest atx=7.5mm and if the muzzle velocity of the bullet is 600m/s at theend of the the 750mm barrel, compute the acceleration of the bulletas it passes the midpoint of the barrel at x=375mm. - The answer given is :104km/s^2.

Explanation / Answer

a = k/x, xo = 7.5 mm, vo = 0 x1 = 750 mm, v1 = 600 m/s, find a when x = 750/2 mm = 0.375 m a = dv/dt = dv/dx dx/dt = v dv/dx = k/x v dv = k dx/x integrate, v2/2 = k lnx + C use xo = 7.5 mm, vo = 0, vo2/2 = klnxo + C so C = -k lnxo v2/2 = k (lnx - lnxo) use x1 = 750 mm, v1 = 600 m/s, v12/2 = k (lnx1 -lnxo) k = v12/[2(lnx1 -lnxo)] = 3.91*104m2/s2 a(x) = k/x = 1.04*105 m/s2 = 104km/s2

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