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)-8.25J
B. (i)1.25J,(ii)-8.25J
C. (i)100J (ii) 8.75J
D. (i)10 J (ii)-8.75 J

Explanation

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 with a mass of 1 g.

### Step-by-Step Solution:

**Step 1: Convert the mass of the raindrop to kilograms.**
- Given mass = 1 g = \(1 \times 10^{-3}\) kg

**Step 2: Convert the height from kilometers to meters.**
- Given height = 1 km = 1000 m

**Step 3: Calculate the work done by the gravitational force.**
- The formula for work done by gravity is:
\[ W_g = mgh \]
where:
- \(m = 1 \times 10^{-3}\) kg (mass of the raindrop)
- \(g = 10 \, \text{m/s}^2\) (acceleration due to gravity)
- \(h = 1000 \, \text{m}\) (height)

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

**Step 4: Calculate the final kinetic energy of the raindrop.**
- The speed when it hits the ground is given as \(v = 50 \, \text{m/s}\).
- The kinetic energy (KE) is given by:
\[ KE = \frac{1}{2} mv^2 \]
- Substituting the values:
\[ KE = \frac{1}{2} \times (1 \times 10^{-3}) \times (50)^2 = \frac{1}{2} \times (1 \times 10^{-3}) \times 2500 = 1.25 \, \text{J} \]

**Step 5: Apply the work-energy theorem to find the work done by the resistive force of air.**
- According to the work-energy theorem:
\[ W_{\text{total}} = KE_{\text{final}} - KE_{\text{initial}} = W_g + W_{\text{air}} \]
- Since the raindrop starts from rest, \(KE_{\text{initial}} = 0\).
- Therefore:
\[ W_{\text{total}} = KE_{\text{final}} = W_g + W_{\text{air}} \]
- Rearranging gives:
\[ W_{\text{air}} = KE_{\text{final}} - W_g \]
- Substituting the values we calculated:
\[ W_{\text{air}} = 1.25 \, \text{J} - 10 \, \text{J} = -8.75 \, \text{J} \]

### Final Answers:
- (i) Work done by gravitational force: **10 J**
- (ii) Work done by resistive force of air: **-8.75 J**