A body has V, 2 2 and 3v in first 2/3 of distance S, second 1/3 of S and third 1/3 of s respectively. Its average speed wil be

Options

A. V
B. 2V
C. `(18)/(11)V`
D. `(11)/(18)V`

Answer

To find the average speed of the body, we will follow these steps:

### Step 1: Determine the total distance traveled
The body travels a total distance \( S \) which is divided into three equal parts:
- First \( \frac{1}{3}S \)
- Second \( \frac{1}{3}S \)
- Third \( \frac{1}{3}S \)

**Total Distance \( D \)**:
\[ D = \frac{1}{3}S + \frac{1}{3}S + \frac{1}{3}S = S \]

### Step 2: Calculate the time taken for each segment
We will calculate the time taken to travel each segment using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Velocity}} \]

1. **Time for the first segment** (velocity = \( V \)):
\[ T_1 = \frac{\frac{1}{3}S}{V} = \frac{S}{3V} \]

2. **Time for the second segment** (velocity = \( 2V \)):
\[ T_2 = \frac{\frac{1}{3}S}{2V} = \frac{S}{6V} \]

3. **Time for the third segment** (velocity = \( 3V \)):
\[ T_3 = \frac{\frac{1}{3}S}{3V} = \frac{S}{9V} \]

### Step 3: Calculate the total time taken
Now, we sum up the time taken for each segment:
\[ T = T_1 + T_2 + T_3 = \frac{S}{3V} + \frac{S}{6V} + \frac{S}{9V} \]

To add these fractions, we need a common denominator. The least common multiple of 3, 6, and 9 is 18. Thus, we rewrite each term:
\[ T = \frac{6S}{18V} + \frac{3S}{18V} + \frac{2S}{18V} = \frac{(6 + 3 + 2)S}{18V} = \frac{11S}{18V} \]

### Step 4: Calculate the average speed
The average speed \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{T} \]
Substituting the values we calculated:
\[ V_{avg} = \frac{S}{\frac{11S}{18V}} = S \cdot \frac{18V}{11S} = \frac{18V}{11} \]

### Final Answer
The average speed of the body is:
\[ \boxed{\frac{18V}{11}} \]