Consider a drop of rain water having mass 1 g falling from a height of `1 km`. It hits the ground with a speed of `50 m//s` Take `g` constant with a volume `10 m//s^(2)`. The work done by the
(i) gravitational force and the
(ii) resistive force of air is :

Options

A. (i)-10J, (ii) -8.25 J
B. (i)1.25 J, (ii) -8.25 J
C. (i)100 J, (ii) -8.25 J
D. (i)10J, (ii) -8.25 J

Answer

To solve the problem, we need to calculate the work done by the gravitational force and the work done by the resistive force of air on a raindrop falling from a height of 1 km.

### Given Data:
- Mass of the raindrop, \( m = 1 \, \text{g} = 1 \times 10^{-3} \, \text{kg} \)
- Height from which it falls, \( h = 1 \, \text{km} = 1000 \, \text{m} \)
- Speed just before hitting the ground, \( v = 50 \, \text{m/s} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

### (i) Work Done by Gravitational Force:
The work done by the gravitational force can be calculated using the formula:
\[ W_g = mgh \]

Substituting the values:
\[ W_g = (1 \times 10^{-3} \, \text{kg}) \times (10 \, \text{m/s}^2) \times (1000 \, \text{m}) \]
\[ W_g = 1 \times 10^{-3} \times 10 \times 1000 \]
\[ W_g = 10 \, \text{J} \]

### (ii) Work Done by Resistive Force of Air:
To find the work done by the resistive force, we can use the principle of conservation of energy. The work done by the resistive force is equal to the change in mechanical energy of the raindrop.

1. **Calculate the initial potential energy (PE)** when the raindrop is at height \( h \):
\[ PE = mgh = (1 \times 10^{-3} \, \text{kg}) \times (10 \, \text{m/s}^2) \times (1000 \, \text{m}) = 10 \, \text{J} \]

2. **Calculate the final kinetic energy (KE)** just before it hits the ground:
\[ KE = \frac{1}{2} mv^2 = \frac{1}{2} \times (1 \times 10^{-3} \, \text{kg}) \times (50 \, \text{m/s})^2 \]
\[ KE = \frac{1}{2} \times (1 \times 10^{-3}) \times 2500 = 1.25 \, \text{J} \]

3. **Calculate the work done by the resistive force (W_r)**:
\[ W_r = KE - PE \]
\[ W_r = 1.25 \, \text{J} - 10 \, \text{J} = -8.75 \, \text{J} \]

### Final Answers:
- Work done by gravitational force, \( W_g = 10 \, \text{J} \)
- Work done by resistive force of air, \( W_r = -8.75 \, \text{J} \)