Rain drops each of mass m falling from rest in air experience an upward force given by f= - bv where b is a constant and v is velocity of the drop. Using work energy theorem, derive an expression for
(A) `v=f(t)` i.e. velocity function of time.
(B) `(dE)/(dt)=f(t)` where E = Total mechanical energy of the rain drop
( C) Draw the graphs `(dE)/(dt), (dU)/(dt) and P_("viscous")` as a function of time U is potential energy and P represent power ?

Explanation

To solve the problem step by step, we will derive the required expressions using the work-energy theorem and analyze the graphs for the given quantities.

### Step 1: Deriving the Velocity Function of Time (v = f(t))

1. **Identify the forces acting on the raindrop**:
- The downward gravitational force: \( F_g = mg \)
- The upward viscous force: \( F_v = -bv \)

...