A raindrop of mass `1g` falling from a height of `1km` hits is the ground with a speed of `50 ms^(-1)`. Which of the following statements is correct? `("Taking" g= 10ms^(-2))`.

Options

A. The loss of potential energy of the drop is 10 J
B. The gain in kinetic energy of the drop is 1.25 J
C. The gain in kinetic energy of the drop is not equal to the loss of potential energy of the drop.
D. All of these

Explanation

To solve the problem, we will calculate the loss of potential energy and the gain in kinetic energy of the raindrop.

### Step 1: Calculate the Loss of Potential Energy (PE)
The formula for potential energy is given by:

\[ PE = mgh \]

Where:
- \( m \) = mass of the raindrop = 1 g = \( \frac{1}{1000} \) kg
- \( g \) = acceleration due to gravity = 10 m/s²
- \( h \) = height from which the raindrop falls = 1 km = 1000 m

Now substituting the values:

\[ PE = \left(\frac{1}{1000} \, \text{kg}\right) \times (10 \, \text{m/s}^2) \times (1000 \, \text{m}) \]

Calculating this gives:

\[ PE = \frac{1}{1000} \times 10 \times 1000 = 10 \, \text{Joules} \]

### Step 2: Calculate the Gain in Kinetic Energy (KE)
The formula for kinetic energy is given by:

\[ KE = \frac{1}{2} mv^2 \]

Where:
- \( m \) = mass of the raindrop = \( \frac{1}{1000} \) kg
- \( v \) = speed of the raindrop when it hits the ground = 50 m/s

Now substituting the values:

\[ KE = \frac{1}{2} \times \left(\frac{1}{1000} \, \text{kg}\right) \times (50 \, \text{m/s})^2 \]

Calculating this gives:

\[ KE = \frac{1}{2} \times \frac{1}{1000} \times 2500 = \frac{1250}{1000} = 1.25 \, \text{Joules} \]

### Step 3: Compare the Loss of Potential Energy and Gain in Kinetic Energy
- Loss of potential energy = 10 Joules
- Gain in kinetic energy = 1.25 Joules

From our calculations, we can see that:

\[ \text{Loss of PE} \neq \text{Gain of KE} \]

### Conclusion
Since the gain in kinetic energy is not equal to the loss of potential energy, this implies that some energy is lost due to factors such as air resistance or viscous drag. Therefore, the correct statement is that the gain in kinetic energy is not equal to the loss of potential energy.

### Final Answer
The correct statement is: "The gain in kinetic energy is not equal to the loss of potential energy."

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