A body travelling along a straight line , one thired of the total distance with a velocity ` 4 ms^(-1)`. The remaining part of the distance was covered with a velocity ` 2 ms^(-1)` for half the time and with velocity ` 6 ms^(-1)` for the other half of time . What is the mean velocity averaged over te whle time of motin ?

Options

A. `5m//s`
B. `4m//s`
C. `4.5m//s`
D. `3.5m//s`

Answer

To solve the problem step by step, let's break it down into manageable parts:

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

### Step 2: Calculate the distance covered at the first velocity
The body travels one-third of the total distance with a velocity of \( 4 \, \text{m/s} \). Thus, the distance covered in this part is:
\[ \text{Distance}_1 = \frac{d}{3} \]

### Step 3: Calculate the time taken for the first part
Using the formula for time, \( \text{time} = \frac{\text{distance}}{\text{velocity}} \), the time taken for this part is:
\[ t_1 = \frac{\text{Distance}_1}{\text{Velocity}_1} = \frac{\frac{d}{3}}{4} = \frac{d}{12} \, \text{s} \]

### Step 4: Calculate the remaining distance
The remaining distance after the first part is:
\[ \text{Remaining Distance} = d - \frac{d}{3} = \frac{2d}{3} \]

### Step 5: Divide the remaining distance into two parts
The remaining distance is covered in two equal halves, where:
- The first half is covered at \( 2 \, \text{m/s} \)
- The second half is covered at \( 6 \, \text{m/s} \)

### Step 6: Set up the time equations for the second part
Let \( t_2 \) be the time taken for each half. The total distance covered in this part can be expressed as:
\[ \text{Distance}_2 = 2 \cdot (2 \, \text{m/s} \cdot t_2) + 2 \cdot (6 \, \text{m/s} \cdot t_2) = 2t_2 + 6t_2 = 8t_2 \]
Setting this equal to the remaining distance:
\[ 8t_2 = \frac{2d}{3} \]

### Step 7: Solve for \( t_2 \)
Rearranging the equation gives:
\[ t_2 = \frac{2d}{3 \cdot 8} = \frac{d}{12} \, \text{s} \]

### Step 8: Calculate total time taken
The total time taken for the entire journey is:
\[ \text{Total time} = t_1 + 2t_2 = \frac{d}{12} + 2 \cdot \frac{d}{12} = \frac{d}{12} + \frac{2d}{12} = \frac{3d}{12} = \frac{d}{4} \, \text{s} \]

### Step 9: Calculate mean velocity
Mean velocity is defined as:
\[ \text{Mean Velocity} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{d}{\frac{d}{4}} = 4 \, \text{m/s} \]

### Final Answer
The mean velocity averaged over the whole time of motion is:
\[ \boxed{4 \, \text{m/s}} \]

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