A particle moving in a straight line covers half the distance with speed of 3 m/s. The other half of the distance covered in two equal time intervals with speed of 4.5 m/s and 7.5 m/s respectively. The average speed of the particle during this motion is :-

Options

A. 4.0 m/s
B. 5.0 m/s
C. 5.5 m/s
D. 4.8 m/s

Answer

To solve the problem of finding the average speed of a particle that covers half the distance at different speeds, we can follow these steps:

### Step 1: Define the total distance
Let the total distance covered by the particle be \( D \). Therefore, half the distance is:
\[ \text{Half distance} = \frac{D}{2} \]

### Step 2: Calculate time taken for the first half
The particle covers the first half of the distance \( \frac{D}{2} \) at a speed of \( 3 \, \text{m/s} \). The time taken \( t_1 \) for this half is given by:
\[ t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{D}{2}}{3} = \frac{D}{6} \, \text{s} \]

### Step 3: Calculate time taken for the second half
The second half of the distance \( \frac{D}{2} \) is covered in two equal time intervals with speeds of \( 4.5 \, \text{m/s} \) and \( 7.5 \, \text{m/s} \). Let the time for each interval be \( t \).

The distance covered in the first interval is:
\[ \text{Distance}_1 = \text{Speed}_1 \times t = 4.5t \]
The distance covered in the second interval is:
\[ \text{Distance}_2 = \text{Speed}_2 \times t = 7.5t \]
The total distance for the second half is:
\[ \text{Distance}_1 + \text{Distance}_2 = 4.5t + 7.5t = 12t \]
Since this distance equals \( \frac{D}{2} \), we can set up the equation:
\[ 12t = \frac{D}{2} \]
Solving for \( t \):
\[ t = \frac{D}{24} \, \text{s} \]

### Step 4: Calculate total time taken
The total time \( T \) taken for the entire journey is the sum of the times for both halves:
\[ T = t_1 + 2t = \frac{D}{6} + 2 \times \frac{D}{24} \]
Calculating \( 2 \times \frac{D}{24} \):
\[ 2 \times \frac{D}{24} = \frac{D}{12} \]
Now, adding both times:
\[ T = \frac{D}{6} + \frac{D}{12} \]
To add these fractions, we need a common denominator:
\[ \frac{D}{6} = \frac{2D}{12} \]
So,
\[ T = \frac{2D}{12} + \frac{D}{12} = \frac{3D}{12} = \frac{D}{4} \, \text{s} \]

### Step 5: Calculate average speed
The average speed \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{D}{4}} = 4 \, \text{m/s} \]

### Final Answer
Thus, the average speed of the particle during this motion is:
\[ \boxed{4 \, \text{m/s}} \]