A particle moving in a straight line covers half the distance with speed of `3 m//s`. The other half of the distance is 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`

Explanation

To solve the problem step by step, we will break down the motion of the particle into two parts and calculate the average speed.

### Step 1: Define the total distance
Let the total distance traveled by the particle be \( x \). According to the problem, the particle covers half of this distance, which is \( \frac{x}{2} \), with a speed of \( 3 \, \text{m/s} \).

### Step 2: Calculate the time taken for the first half
The time taken to cover the first half of the distance can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
So, for the first half:
\[ t_1 = \frac{\frac{x}{2}}{3} = \frac{x}{6} \, \text{seconds} \]

### Step 3: Calculate the time taken for the second half
The second half of the distance \( \frac{x}{2} \) 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 \( t_2 \).

The distance covered in the first interval is:
\[ \text{Distance}_1 = 4.5 \times t_2 \]
The distance covered in the second interval is:
\[ \text{Distance}_2 = 7.5 \times t_2 \]
Since these two distances together equal half of the total distance:
\[ 4.5 t_2 + 7.5 t_2 = \frac{x}{2} \]
Combining the terms gives:
\[ 12 t_2 = \frac{x}{2} \]
Solving for \( t_2 \):
\[ t_2 = \frac{x}{24} \, \text{seconds} \]

### Step 4: Calculate the total time taken
The total time taken \( T \) is the sum of the time for the first half and the time for the second half (which consists of two intervals):
\[ T = t_1 + 2t_2 = \frac{x}{6} + 2 \left( \frac{x}{24} \right) \]
Calculating \( 2t_2 \):
\[ 2t_2 = 2 \times \frac{x}{24} = \frac{x}{12} \]
Now, substituting back into the total time equation:
\[ T = \frac{x}{6} + \frac{x}{12} \]
To add these fractions, we need a common denominator (which is 12):
\[ T = \frac{2x}{12} + \frac{x}{12} = \frac{3x}{12} = \frac{x}{4} \, \text{seconds} \]

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

### Final Answer
The average speed of the particle during this motion is \( 4 \, \text{m/s} \).

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