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Projectile Motion with Air Drag

I need help with these two questions:
2.12. A gun is fired straight up. Assuming that the air drag on the bullet varies quadratically with speed, show that the speed varies with height according to the equations:
v^2 = Ae^-2kx - g/k (upward motion)
v^2 = g/k - Be^2kx (downward motion)
in which A and B are constants of integration, g is the acceleration of gravity, and k =c2/m where c2 is the drag constant and m is the mass of the bullet. (Note: x is measured positive upward, and the gravitational force is assumed to be constant).

2.18. The force acting on a particle of mass m is given by:
F = kvx
which k is a positive constant. The particle passes through the origin with speed vo at time t=0. Find x as a function of t.


Solution Summary

This solution contains detailed step-by-step calculations to show that the speed varies with height of the bullet and also the displacement of the particle passing through the origin.