Putu N.

show that the speed varies with height according to the equation below

A gun is fired straight up. Assuming that the air drag on the bullets varies quadratically with speed,

(1) show that the speed varies with height according to the equations:
•  v2 = Ae-2kx - 8/k (upward motion)
•  v2= 8/k - Be2kx (downward motion)
in which A and B are constant of integration,
g = acceleration of gravicity;
k= C2/m, C is the drag constant;
m= mass of bullet.
{Note: x is measured positive upward, and the gravitational force is assumed to be constant.

(2) when the bullets hits the ground on its return, the speed will be equal to expression:
VoV1/(Vo2+V12)1/2
in which Vo is the initial upward speed and
Vt = (mg/C2)1/2 = terminal speed = (g/k)1/2

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