Step 1: Work-Energy Theorem

The change in kinetic energy of the raindrop is equal to the total work done by all forces acting on it (Gravity and Viscous Drag).

$$W_{gravity} + W_{drag} = \Delta K$$

Step 2: Calculate Work Components

• Work done by gravity: $W_g = mgh$ (Positive, as displacement is downward).
• Change in Kinetic Energy: $\Delta K = K_f – K_i = \frac{1}{2} m v_T^2 – 0$.
• Work done by drag: Let the magnitude of the average viscous force be $F$. Since drag opposes motion, the work done is negative: $W_{drag} = – F \cdot h$.

Step 3: Solve for Average Force $F$

Substituting the values into the Work-Energy equation:

$$mgh – F \cdot h = \frac{1}{2} m v_T^2$$

Rearranging to solve for $F$:

$$F \cdot h = mgh – \frac{1}{2} m v_T^2$$ $$F = mg – \frac{m v_T^2}{2h}$$ $$F = m \left( g – \frac{v_T^2}{2h} \right)$$