Projectile Motion with Air Resistance (Position function?)

[gman07]

Homework Statement

There is no real problem statement for this problem, except that I'm trying to figure out an equation for the position of a projectile with air resistance. It's for a physical model of a toy gun, fired at 0° from the horizontal at a height of 1m. I'm considering only air resistance due to motion in the x direction.

Homework Equations

Using F=k*v^2 and k=0.5*Cd*A*p. I set up a differential equation (dv/dt = -(k/m)v^2) and solved with the initial condition V(0)= v (muzzle velocity) to get V(t) = (m/k)*(1/(t+(m/(v*k)))). In this, m is the projectile mass, t is time, v is muzzle velocity.

However, I'm having trouble integrating this with respect to t to find position as a function of time. I think some of the problem may lie in the fact that the function I have for velocity integrates the initial velocity into it, so that V(0) = v.

If nothing else, I could estimate it in a calculus-less manner.

 

[NateD]

I'm just having trouble figuring out what the equation should be.The Attempt at a SolutionI think the equation I'm trying to find is x(t) = v0*t - (k/m)*ln(t+(m/(v0*k))) + x0, where x0 = 0. However, this equation doesn't seem to match up with the results of my model, which follows more closely with x(t) = v0*t - (1/2)*(k/m)*t^2 +x0.