1. The problem statement, all variables and given/known data A gun is fired straight up. Assuming that the air drag on the bullet varies quadratically with speed. Show that 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, k=cm, where c 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. Relevant equations m dv/dt = mg - cv^2 = mg(1 - cv^2/mg) .'. dv/dt = g(1 - v^2/v(sub t)^2) where v(sub t) is the terminal speed, sqrt(mg/c) dv/dt = dv/dx dx/dt = 1/2 dv^2/dx So now we can write the equation as: dv^2/dx = 2g(1 - v^2/v(sub t)^2) 3. The attempt at a solution I have gotten as far as to say: For downward motion: v^2 = g/k(1exp^[2kx]) + v(naught)^2.exp^[2kx]Likewise for upward, we get: v^2 = g/k(1-exp^[-2kx]) + v(naught)^2.exp^[-2kx] I just don't know how to relate these to the original equation to show that speed varies with height.