It is a common observation that rain clouds can be at about 1 km altitude above the ground. If a rain drop falls from such a height freely under gravity, then what will be its speed in km `h^(-1)`?
(Take g = `10 m s^(-2)`)

Options

A. a. 510
B. b. 610
C. c. 710
D. d. 910

Explanation

To solve the problem of finding the speed of a raindrop falling from a height of 1 km under the influence of gravity, we can follow these steps:

### Step 1: Understand the given data
- Height (h) = 1 km = 1000 m
- Initial velocity (u) = 0 m/s (since the raindrop starts from rest)
- Acceleration due to gravity (g) = 10 m/s²

### Step 2: Use the third equation of motion
The third equation of motion relates the final velocity (V), initial velocity (u), acceleration (g), and distance (h) as follows:
\[ V^2 = u^2 + 2gh \]

### Step 3: Substitute the known values
Substituting the values into the equation:
\[ V^2 = 0^2 + 2 \times 10 \, \text{m/s}^2 \times 1000 \, \text{m} \]
\[ V^2 = 0 + 20000 \]
\[ V^2 = 20000 \]

### Step 4: Calculate the final velocity (V)
Taking the square root of both sides:
\[ V = \sqrt{20000} \]
\[ V = 100\sqrt{2} \, \text{m/s} \]

### Step 5: Convert the velocity from m/s to km/h
To convert from meters per second to kilometers per hour, we use the conversion factor \( \frac{18}{5} \):
\[ V = 100\sqrt{2} \times \frac{18}{5} \]

### Step 6: Calculate the numerical value
First, calculate \( \sqrt{2} \approx 1.414 \):
\[ V \approx 100 \times 1.414 \times \frac{18}{5} \]
\[ V \approx 100 \times 1.414 \times 3.6 \]
\[ V \approx 100 \times 5.0944 \]
\[ V \approx 509.44 \, \text{km/h} \]

### Step 7: Round off the value
Rounding off gives us approximately:
\[ V \approx 510 \, \text{km/h} \]

### Conclusion
Thus, the speed of the raindrop when it reaches the ground is approximately **510 km/h**.

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