A raindrop of mass 1 g falling from a height of 1 km hits the ground with a speed of `50 m s^(-1)`. If the resistive force is proportional to the speed of the drop, then the work done by the resistive force is (Take `g = 10 m s^(-2)`)

Options

A. 10 J
B. -10 J
C. 8.75J
D. `-8.75 J`

Explanation

To solve the problem, we need to find the work done by the resistive force acting on a raindrop as it falls from a height of 1 km and hits the ground with a speed of 50 m/s. We will use the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy.

### Step-by-Step Solution:

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

2. **Calculate the Gravitational Potential Energy (GPE) at the height:**
\[ \text{GPE} = mgh = (1 \times 10^{-3} \, \text{kg}) \times (10 \, \text{m/s}^2) \times (1000 \, \text{m}) = 10 \, \text{J} \]

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

4. **Use the Work-Energy Theorem:**
According to the work-energy theorem:
\[ \text{Work done by all forces} = \text{Change in Kinetic Energy} \]
The work done by the gravitational force (Wg) and the work done by the resistive force (Wr) can be expressed as:
\[ Wg + Wr = \Delta KE \]
Rearranging gives:
\[ Wr = \Delta KE - Wg \]

5. **Calculate the Change in Kinetic Energy:**
\[ \Delta KE = KE - 0 = 1.25 \, \text{J} \]

6. **Substitute the values to find Work done by the resistive force:**
\[ Wr = 1.25 \, \text{J} - 10 \, \text{J} = -8.75 \, \text{J} \]

### Final Answer:
The work done by the resistive force is \( -8.75 \, \text{J} \).