A particle travels 1st half of its total distance with a speed `V_(1)`, next one fourth with `V_(2)` and rest with `V_(3)`. Its average speed for entire one dimensional motion in terms of `V_(1), V_(2) and V_(3)` is

Options

A. `(4V_(1)V_(2)V_(3))/(2V_(1)+V_(2) + V_(3))`
B. `(4V_(1)V_(2)V_(3))/(2V_(2)V_(3) + V_(1)V_(2) + V_(1)V_(3))`
C. `(2V_(1) + V_(2) + V_(3))/(4)`
D. None of these

Answer

To find the average speed of a particle that travels different segments of a total distance with varying speeds, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Distance**:
Let the total distance traveled by the particle be \( s \).

2. **Determine Distances for Each Segment**:
- The first half of the distance is \( \frac{s}{2} \) and is traveled at speed \( V_1 \).
- The next one-fourth of the distance is \( \frac{s}{4} \) and is traveled at speed \( V_2 \).
- The remaining distance, which is also \( \frac{s}{4} \) (since \( s - \frac{s}{2} - \frac{s}{4} = \frac{s}{4} \)), is traveled at speed \( V_3 \).

3. **Calculate Time Taken for Each Segment**:
- Time taken for the first segment \( t_1 \):
\[ t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{s}{2}}{V_1} = \frac{s}{2V_1} \]
- Time taken for the second segment \( t_2 \):
\[ t_2 = \frac{\frac{s}{4}}{V_2} = \frac{s}{4V_2} \]
- Time taken for the third segment \( t_3 \):
\[ t_3 = \frac{\frac{s}{4}}{V_3} = \frac{s}{4V_3} \]

4. **Calculate Total Time Taken**:
The total time \( T \) taken for the entire journey is the sum of the times for each segment:
\[ T = t_1 + t_2 + t_3 = \frac{s}{2V_1} + \frac{s}{4V_2} + \frac{s}{4V_3} \]

5. **Factor Out \( s \)**:
\[ T = s \left( \frac{1}{2V_1} + \frac{1}{4V_2} + \frac{1}{4V_3} \right) \]

6. **Calculate Average Speed**:
The average speed \( V_{\text{avg}} \) is defined as the total distance divided by the total time:
\[ V_{\text{avg}} = \frac{s}{T} = \frac{s}{s \left( \frac{1}{2V_1} + \frac{1}{4V_2} + \frac{1}{4V_3} \right)} = \frac{1}{\left( \frac{1}{2V_1} + \frac{1}{4V_2} + \frac{1}{4V_3} \right)} \]

7. **Simplify the Expression**:
To simplify the expression, we can find a common denominator:
\[ V_{\text{avg}} = \frac{1}{\frac{2V_2V_3 + V_1V_3 + V_1V_2}{4V_1V_2V_3}} = \frac{4V_1V_2V_3}{2V_2V_3 + V_1V_3 + V_1V_2} \]

### Final Result:
Thus, the average speed of the particle for the entire motion is:
\[ V_{\text{avg}} = \frac{4V_1V_2V_3}{2V_2V_3 + V_1V_3 + V_1V_2} \]