Imagine we drop an object from a helicopter that is very high above the ground. It turns out that the downward velocity will not approach infinity - instead, the velocity will approach a terminal velocity. The textbook discusses a model that describes the rate of change of the velocity; please read over this example first (see Example 7.3 on p.170). Another model to describe the velocity's rate of change is =9-v, v(0) = 0, where t is time after dropping the object and u is the downward velocity (i.e., v> 0 means the object is travelling downwards, while v < 0 means upwards). For simplicity of the calculations, we have taken the gravitational constant to be g = 9. We will explore two methods to find the terminal velocity. du dt (a) Method 1- Seperation of Variables i. Use separation of variables to find v(t), then use the initial condition to solve for the constant of integration. ii. Compute the terminal velocity by looking at the long term behaviour of your solution v(t) (i.e., find lim v(t)). (b) Method 2- Equilibria i. Find both steady states of this differential equation, and state which one is physically irrelevant. ii. Find the stability of this steady state. Note that this steady state is the terminal velocity. (c) Reflect on which method was easier to find the terminal velocity (for this part, you do not have to hand anything in).