A particle moving in a straight line covers first half of the distance into two equal time intervals with speed of 2 m/s and 4 m/s respectively, and second half with the speed of 5 m/s. The average speed of the particle is

Options

A. 4.75 m/s
B. 4.25 m/s
C. 4.00 m/s
D. 3.75 m/s

Answer

To find the average speed of the particle, we will follow these steps:

### Step 1: Define the total distance
Let the total distance covered by the particle be \( S \). Therefore, the first half of the distance is \( \frac{S}{2} \) and the second half is also \( \frac{S}{2} \).

### Step 2: Calculate the time taken for the first half of the distance
The first half of the distance \( \frac{S}{2} \) is covered in two equal time intervals with speeds of 2 m/s and 4 m/s.

1. **First Interval (O to A)**:
- Distance = \( \frac{S}{4} \)
- Speed = 2 m/s
- Time taken \( t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{S}{4}}{2} = \frac{S}{8} \)

2. **Second Interval (A to B)**:
- Distance = \( \frac{S}{4} \)
- Speed = 4 m/s
- Time taken \( t_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{S}{4}}{4} = \frac{S}{16} \)

### Step 3: Calculate the time taken for the second half of the distance
For the second half of the distance \( \frac{S}{2} \), the speed is 5 m/s.

- Time taken \( t_3 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{S}{2}}{5} = \frac{S}{10} \)

### Step 4: Calculate the total time taken
Now, we can find the total time taken \( T \) by summing up the time intervals:

\[ T = t_1 + t_2 + t_3 = \frac{S}{8} + \frac{S}{16} + \frac{S}{10} \]

### Step 5: Find a common denominator to simplify the total time
To add these fractions, we need a common denominator. The least common multiple (LCM) of 8, 16, and 10 is 80.

1. Convert each term:
- \( \frac{S}{8} = \frac{10S}{80} \)
- \( \frac{S}{16} = \frac{5S}{80} \)
- \( \frac{S}{10} = \frac{8S}{80} \)

2. Now, add them:
\[ T = \frac{10S}{80} + \frac{5S}{80} + \frac{8S}{80} = \frac{(10 + 5 + 8)S}{80} = \frac{23S}{80} \]

### Step 6: Calculate the average speed
The average speed \( V_{avg} \) is given by the formula:

\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{S}{T} \]

Substituting the value of \( T \):

\[ V_{avg} = \frac{S}{\frac{23S}{80}} = \frac{80}{23} \text{ m/s} \]

### Step 7: Calculate the numerical value
Calculating \( \frac{80}{23} \):

\[ V_{avg} \approx 3.48 \text{ m/s} \]

### Final Result
Thus, the average speed of the particle is approximately **3.48 m/s**.

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