A parachutist falls from a balloon at a height of y0 (=4000). k1 represents air resistance in stage one and k2 in stage 2. Air resistance is assumed proportional to velocity.

I've split this problem into two sections - free fall and after the parachute is deployed.

For the first section the initial conditions are v(0) = 0 and I have the formula

and for stage two I have

where td is the time of deploying the parachute.

I've integrated the first stage v(t) to find x(t), but I'm struggling to integrate the second stage. Please help.