A body travelling along a straight line traversed one third of the total distance with a velocity 4m/s. The remaining part of the distance was covered with a velocity 1 m/s for half the time and with velocity 3 m/s for the other half of time. The mean velocity averaged over the whole time of motion is :

Options

A. (a)`2.4 m//s`
B. (b)` 4 m//s`
C. (c)` 1.9 m//s`
D. (d)` 3 m//s`

Explanation

To find the mean velocity of the body over the entire motion, we can break down the problem into steps.

### Step 1: Define the total distance
Let the total distance traveled by the body be \( d \).

### Step 2: Calculate the distance for the first part
The body travels one third of the total distance with a velocity of \( 4 \, \text{m/s} \).
Thus, the distance for the first part is:
\[ d_1 = \frac{d}{3} \]

### Step 3: Calculate the time taken for the first part
The time taken to cover this distance can be calculated using the formula:
\[ t_1 = \frac{\text{distance}}{\text{velocity}} = \frac{d/3}{4} = \frac{d}{12} \, \text{s} \]

### Step 4: Calculate the remaining distance
The remaining distance is:
\[ d_2 = d - d_1 = d - \frac{d}{3} = \frac{2d}{3} \]

### Step 5: Calculate the time taken for the remaining distance
The remaining distance is covered in two halves of time. Let \( t_2 \) be the total time for the remaining distance. The body travels with:
- Velocity \( 1 \, \text{m/s} \) for half the time
- Velocity \( 3 \, \text{m/s} \) for the other half

Let the time for each half be \( t_2/2 \). Then, the distance covered in each half is:
1. Distance covered at \( 1 \, \text{m/s} \):
\[ d_{2a} = 1 \times \frac{t_2}{2} = \frac{t_2}{2} \]
2. Distance covered at \( 3 \, \text{m/s} \):
\[ d_{2b} = 3 \times \frac{t_2}{2} = \frac{3t_2}{2} \]

### Step 6: Set up the equation for the remaining distance
The total distance for the remaining part is:
\[ d_{2a} + d_{2b} = \frac{t_2}{2} + \frac{3t_2}{2} = 2t_2 \]
Setting this equal to the remaining distance:
\[ 2t_2 = \frac{2d}{3} \]
Thus, we can solve for \( t_2 \):
\[ t_2 = \frac{d}{3} \]

### Step 7: Calculate total time
The total time \( T \) for the entire journey is:
\[ T = t_1 + t_2 = \frac{d}{12} + \frac{d}{3} \]
To add these, convert \( \frac{d}{3} \) to have a common denominator:
\[ \frac{d}{3} = \frac{4d}{12} \]
Thus,
\[ T = \frac{d}{12} + \frac{4d}{12} = \frac{5d}{12} \]

### Step 8: Calculate the mean velocity
The mean velocity \( V_{avg} \) is given by the total distance divided by the total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{d}{\frac{5d}{12}} = \frac{d \times 12}{5d} = \frac{12}{5} = 2.4 \, \text{m/s} \]

### Final Answer
The mean velocity averaged over the whole time of motion is \( 2.4 \, \text{m/s} \).