Using Differential Equations to model a weather balloon flight

[andersonty319]

I'm supposed to created a differential equation for the altitude of a weather balloon. Here's what I have: at time 0, altitude is 0 m, and initial velocity is 0 m/s. The balloon has a mass of 2 kg and exerts an upward force of 26 N. There is a drag force that is approximately 1/5 of the instantaneous velocity. After several seconds, a dart gun is fired at the balloon. It hits and gives the balloon a small jolt at time t = 7. Even though the balloon is deflated the amount of drag does not change.

 

Ok, so here's an equation modeling velocity:

 

2*dv/dt = -9.8(2) + 26 - 26u(t-7) + delta(t-7) - .2v

 

(the u is a step function and the "delta" is for the Dirac function.)

 

I understand the different parts of the velocity equation. The 2*dv/dt represents the mass times acceleration, the -9.8(2) represents the weight force, the 26 represents the lift, the step function "shuts off" the lift at 7 seconds, the Dirac function models the impulse, and the -.2v models the drag.

 

I'm stuck on how to create an equation for the altitude in terms of t. Any advice on getting started?

Thanks! You guys are the best.