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. `2.4 m//s`
B. ` 4 m//s`
C. ` 1.9 m//s`
D. ` 3 m//s`

Explanation

To solve the problem, we need to find the mean velocity of a body that travels along a straight line with varying velocities. Let's break it down step by step.

### Step 1: Define the total distance
Let the total distance be \( D \). According to the problem, the body travels one third of the total distance with a velocity of \( 4 \, \text{m/s} \).

### Step 2: Calculate the distance for the first part
The distance covered in 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:
\[ t_1 = \frac{d_1}{v_1} = \frac{\frac{D}{3}}{4} = \frac{D}{12} \]

### Step 4: Calculate the remaining distance
The remaining distance \( d_2 \) 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 parts:
- The first half of the time is spent covering this distance at \( 1 \, \text{m/s} \).
- The second half of the time is spent covering this distance at \( 3 \, \text{m/s} \).

Let the total time for the remaining distance be \( t_2 \). Thus, the time spent at each velocity is \( \frac{t_2}{2} \).

### Step 6: Calculate the distance covered at each velocity
1. Distance covered at \( 1 \, \text{m/s} \):
\[ d_{2,1} = v_1 \cdot \frac{t_2}{2} = 1 \cdot \frac{t_2}{2} = \frac{t_2}{2} \]

2. Distance covered at \( 3 \, \text{m/s} \):
\[ d_{2,2} = v_2 \cdot \frac{t_2}{2} = 3 \cdot \frac{t_2}{2} = \frac{3t_2}{2} \]

### Step 7: Set up the equation for the remaining distance
The total distance for the remaining part is:
\[ d_{2,1} + d_{2,2} = \frac{t_2}{2} + \frac{3t_2}{2} = 2D/3 \]
This gives us:
\[ \frac{t_2}{2} + \frac{3t_2}{2} = \frac{2D}{3} \]
\[ 2t_2 = \frac{2D}{3} \implies t_2 = \frac{D}{3} \]

### Step 8: Calculate the total time
The total time \( t \) is:
\[ t = t_1 + t_2 = \frac{D}{12} + \frac{D}{3} \]
To add these, we 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 9: Calculate the mean velocity
The mean velocity \( v_{mean} \) is given by:
\[ v_{mean} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{5D}{12}} = \frac{12}{5} \, \text{m/s} \]

### Final Answer
The mean velocity averaged over the whole time of motion is:
\[ \boxed{\frac{12}{5} \, \text{m/s}} \]