Differential equation problem

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A 500kg balloon gondola has zero velocity at time t = 0 seconds when it tears loose from the balloon. As it falls it is subject to the force of gravity and to the force of air resistance, which is 1/50 v^2 when the gondola has velocity v metres per second.

a. Find the downward velocity as a function of t and the terminal velocity.
b. If the gondola was 1500m above the ground when it broke loose, how long would it take to reach the ground and what is its velocity at this time?

thanks in advance
 
forces acting on the gondola (using g = 10 m/s<sup>2</sup>) ...
weight = mg = 5000 N down, air resistance = v<sup>2</sup>/50 up

F<sub>net</sub> = 5000 - v<sup>2</sup>/50
ma = 5000 - v<sup>2</sup>/50
500a = 5000 - v<sup>2</sup>/50
a = dv/dt = 10 - v<sup>2</sup>/25000

dv/dt = (250000 - v<sup>2</sup>)/25000
dv/(250000 - v<sup>2</sup>) = 25000 dt
dv/[(500 - v)(500 + v)] = 25000 dt

now, use the method of partial fractions to integrate the left side.
 
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