A body travelling along a straight line traversed one-third of the total distance with a velocity `v_1` . The remaining part of the distance was covered with a velocity `v_2` for half the time and with velocity `v_3` for the other half of time. The mean velocity averaged over the whole time of motion

Options

A. `(3v_(1)(v_(2)+v_(3)))/(2v_(1)+v_(2)+v_(3))`
B. `(3v_(1)(v_(2)+v_(3)))/(4v_(1)+v_(2)+v_(3))`
C. `(v_(1)(v_(2)+v_(3)))/(4v_(1)+v_(2)+v_(3))`
D. `(v_(1)(v_(2)+v_(3)))/(v_(1)+v_(2)+v_(3))`

Explanation

To find the mean velocity of the body traveling along a straight line, we can follow these steps:

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

### Step 2: Calculate the Distance Covered at Velocity \( v_1 \)
The body covers one-third of the total distance with velocity \( v_1 \):
\[ \text{Distance covered with } v_1 = \frac{x}{3} \]

### Step 3: Calculate the Time Taken for the First Part
The time taken to cover this distance at velocity \( v_1 \) is given by:
\[ t_1 = \frac{\text{Distance}}{\text{Velocity}} = \frac{\frac{x}{3}}{v_1} = \frac{x}{3v_1} \]

### Step 4: Calculate the Remaining Distance
The remaining distance to be covered is:
\[ \text{Remaining distance} = x - \frac{x}{3} = \frac{2x}{3} \]

### Step 5: Time Distribution for Remaining Distance
The remaining distance is covered in two halves of time \( t_2 \):
- Half of the time \( t_2 \) is spent traveling at velocity \( v_2 \).
- The other half of the time \( t_2 \) is spent traveling at velocity \( v_3 \).

### Step 6: Express the Total Time for Remaining Distance
The total distance covered in the remaining distance can be expressed as:
\[ \text{Distance} = \text{Distance at } v_2 + \text{Distance at } v_3 \]
\[ \frac{2x}{3} = v_2 t_2 + v_3 t_2 \]
\[ \frac{2x}{3} = t_2 (v_2 + v_3) \]
From this, we can solve for \( t_2 \):
\[ t_2 = \frac{\frac{2x}{3}}{v_2 + v_3} = \frac{2x}{3(v_2 + v_3)} \]

### Step 7: Calculate Total Time
The total time \( T \) for the entire journey is:
\[ T = t_1 + t_2 + t_2 = t_1 + 2t_2 \]
Substituting the values of \( t_1 \) and \( t_2 \):
\[ T = \frac{x}{3v_1} + 2 \left( \frac{2x}{3(v_2 + v_3)} \right) \]
\[ T = \frac{x}{3v_1} + \frac{4x}{3(v_2 + v_3)} \]

### Step 8: Calculate Mean Velocity
The mean velocity \( V_{avg} \) is defined as total distance divided by total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{x}{T} \]
Substituting for \( T \):
\[ V_{avg} = \frac{x}{\frac{x}{3v_1} + \frac{4x}{3(v_2 + v_3)}} \]
Factoring out \( x \):
\[ V_{avg} = \frac{1}{\frac{1}{3v_1} + \frac{4}{3(v_2 + v_3)}} \]
Taking the common denominator:
\[ V_{avg} = \frac{3v_1(v_2 + v_3)}{(v_2 + v_3) + 4v_1} \]
This simplifies to:
\[ V_{avg} = \frac{3v_1v_2 + 3v_1v_3}{4v_1 + v_2 + v_3} \]

### Final Result
Thus, the mean velocity averaged over the whole time of motion is:
\[ V_{avg} = \frac{3v_1v_2 + 3v_1v_3}{4v_1 + v_2 + v_3} \]