Twenty seven identical rain drops colaese to form a big rain drop. If this rain drop travels with terminal velocity of 18 `ms^(-1)`, what is the terminal velocity of each small rain drop?

Options

A. a. 4 `ms^(-1)`
B. b. 2 `ms^(-1)`
C. c. 6 `ms^(-1)`
D. d. 10 `ms^(-1)`

Answer

To solve the problem, we need to find the terminal velocity of each small raindrop given that 27 identical raindrops coalesce to form a larger raindrop that travels with a terminal velocity of 18 m/s.

### Step-by-Step Solution:

1. **Understand the relationship between the volumes of the small and big raindrops**:
- The volume of one small raindrop (V_small) is given by the formula:
\[ V_{\text{small}} = \frac{4}{3} \pi r_1^3 \]
- Since there are 27 small raindrops, the total volume of the small raindrops is:
\[ V_{\text{total small}} = 27 \times V_{\text{small}} = 27 \times \frac{4}{3} \pi r_1^3 = 36 \pi r_1^3 \]
- The volume of the big raindrop (V_big) is:
\[ V_{\text{big}} = \frac{4}{3} \pi r_2^3 \]

2. **Set the total volume of the small raindrops equal to the volume of the big raindrop**:
\[ 36 \pi r_1^3 = \frac{4}{3} \pi r_2^3 \]
- Cancel out \(\pi\) from both sides:
\[ 36 r_1^3 = \frac{4}{3} r_2^3 \]
- Multiply both sides by 3:
\[ 108 r_1^3 = 4 r_2^3 \]
- Rearranging gives:
\[ r_2^3 = 27 r_1^3 \]
- Taking the cube root:
\[ r_2 = 3 r_1 \]

3. **Relate terminal velocities to the radii of the raindrops**:
- The terminal velocity (V_t) is proportional to the square of the radius (r):
\[ V_t \propto r^2 \]
- Therefore, we can write the relationship between the terminal velocities of the small and big raindrops:
\[ \frac{V_{t1}}{V_{t2}} = \frac{r_1^2}{r_2^2} \]
- Substituting \(r_2 = 3 r_1\):
\[ \frac{V_{t1}}{V_{t2}} = \frac{r_1^2}{(3 r_1)^2} = \frac{r_1^2}{9 r_1^2} = \frac{1}{9} \]

4. **Substituting the known terminal velocity of the big raindrop**:
- Given that \(V_{t2} = 18 \, \text{m/s}\):
\[ V_{t1} = \frac{1}{9} V_{t2} = \frac{1}{9} \times 18 = 2 \, \text{m/s} \]

5. **Conclusion**:
- The terminal velocity of each small raindrop is:
\[ V_{t1} = 2 \, \text{m/s} \]

### Final Answer:
The terminal velocity of each small raindrop is **2 m/s**.