A particle travels half of the time with constant speed `2 m//s,` In remaining half of the time it travels, `1/4 th` distance with constant speed of `4 m//s` and `3/4 th` distance with `6 m//s.` Find average speed during the complete journey.

Explanation

To find the average speed of the particle during its complete journey, we can follow these steps:

### Step 1: Understand the problem
The particle travels half of the total time with a speed of 2 m/s and the other half of the time it travels part of the distance with different speeds. We need to calculate the total distance traveled and the total time taken to find the average speed.

### Step 2: Define total time
Let the total time of the journey be \( T \). According to the problem, the particle travels for half of this time, which is \( \frac{T}{2} \), at a speed of 2 m/s.

### Step 3: Calculate distance for the first half
The distance traveled during the first half of the time (let's call it \( d_1 \)) can be calculated as:
\[ d_1 = \text{speed} \times \text{time} = 2 \, \text{m/s} \times \frac{T}{2} = T \, \text{m} \]

### Step 4: Analyze the second half of the journey
In the remaining half of the time \( \frac{T}{2} \), the particle covers \( \frac{1}{4} \) of the total distance \( d \) at a speed of 4 m/s and \( \frac{3}{4} \) of the total distance at a speed of 6 m/s.

Let the total distance \( d \) be the sum of \( d_1 \) and \( d \):
\[ d = d_1 + d \]

### Step 5: Calculate time taken for the second half
1. **Time taken to travel \( \frac{1}{4} d \) at 4 m/s:**
\[ t_2 = \frac{\frac{1}{4} d}{4} = \frac{d}{16} \]

2. **Time taken to travel \( \frac{3}{4} d \) at 6 m/s:**
\[ t_3 = \frac{\frac{3}{4} d}{6} = \frac{3d}{24} = \frac{d}{8} \]

### Step 6: Total time for the second half
The total time for the second half of the journey is:
\[ t_2 + t_3 = \frac{d}{16} + \frac{d}{8} = \frac{d}{16} + \frac{2d}{16} = \frac{3d}{16} \]

### Step 7: Set the equation for total time
Since the second half of the journey also takes \( \frac{T}{2} \):
\[ \frac{3d}{16} = \frac{T}{2} \]
From this, we can express \( d \) in terms of \( T \):
\[ d = \frac{16T}{3} \]

### Step 8: Calculate total distance
The total distance \( D \) is:
\[ D = d_1 + d = T + \frac{16T}{3} = \frac{3T}{3} + \frac{16T}{3} = \frac{19T}{3} \]

### Step 9: Calculate average speed
The average speed \( V_{avg} \) is given by:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{T} = \frac{\frac{19T}{3}}{T} = \frac{19}{3} \, \text{m/s} \]

### Final Answer
The average speed during the complete journey is:
\[ \boxed{\frac{19}{3} \, \text{m/s}} \]