A particle travels half the distance of a straight journey with a speed 5 m/s. The remaining part of the distance is covered with speed 6 m/s for half the remaining time, and with speed 4 m/s for the other half of the remaining time. The average speed of the particle is

Options

A. 3 m/s
B. 4 m/s
C. `(3)/(4)` m/s
D. 5 m/s

Answer

To solve the problem step by step, we will calculate the average speed of the particle based on the given conditions.

### Step 1: Define the total distance
Let the total distance of the journey from point A to point B be \( D \). According to the problem, the particle travels half of this distance first, which means the distance from A to C (the midpoint) is \( \frac{D}{2} \).

### Step 2: Calculate the time taken for the first half of the journey
The speed for the first half of the journey (from A to C) is given as 5 m/s. The time taken to cover this distance can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{D}{2}}{5} = \frac{D}{10} \]
Let this time be \( T' \). Therefore, \( T' = \frac{D}{10} \).

### Step 3: Analyze the second half of the journey
The remaining distance from C to B is also \( \frac{D}{2} \). This distance is covered in two parts:
- The first half of the remaining time is traveled at a speed of 6 m/s.
- The second half of the remaining time is traveled at a speed of 4 m/s.

Let the total time taken to cover the second half of the journey be \( T \). Thus, the time for each part is:
- First half time: \( \frac{T}{2} \)
- Second half time: \( \frac{T}{2} \)

### Step 4: Calculate the distances covered in the second half
The distance covered in the first half of the remaining time is:
\[ \text{Distance}_1 = \text{Speed} \times \text{Time} = 6 \times \frac{T}{2} = 3T \]
The distance covered in the second half of the remaining time is:
\[ \text{Distance}_2 = \text{Speed} \times \text{Time} = 4 \times \frac{T}{2} = 2T \]
The total distance for the second half of the journey is:
\[ \text{Total Distance}_\text{second half} = 3T + 2T = 5T \]

### Step 5: Calculate the total distance
Now, we can express the total distance \( D \):
\[ D = \frac{D}{2} + \frac{D}{2} = 5T + \frac{D}{10} \]
This gives us the equation:
\[ D = 5T \]

### Step 6: Calculate the total time taken
The total time taken for the entire journey is:
\[ \text{Total Time} = T' + T = \frac{D}{10} + T \]
Substituting \( T \) from the previous equation \( T = \frac{D}{5} \):
\[ \text{Total Time} = \frac{D}{10} + \frac{D}{5} = \frac{D}{10} + \frac{2D}{10} = \frac{3D}{10} \]

### Step 7: Calculate the average speed
The average speed \( V_{\text{avg}} \) is given by:
\[ V_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{3D}{10}} = \frac{D \times 10}{3D} = \frac{10}{3} \text{ m/s} \]

### Final Answer
The average speed of the particle is \( \frac{10}{3} \) m/s.