A body moves in straight line and covers first half of the distance with speed 5 m/s and second half in two equal halves with speed 4 m/s and 3 m/s respectively. The average velocity of the body is nearly equal to

Options

A. 4 m/s
B. 3 m/s
C. 2 m/s
D. 2.5 m/s

Answer

To find the average velocity of the body, we can follow these steps:

### Step 1: Define the total distance
Let the total distance traveled by the body be \( d \). The body covers the first half of the distance \( \frac{d}{2} \) with a speed of \( 5 \, \text{m/s} \).

### Step 2: Calculate time for the first half
The time taken to cover the first half of the distance can be calculated using the formula:
\[ t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{d}{2}}{5} = \frac{d}{10} \]

### Step 3: Define the second half of the distance
The second half of the distance \( \frac{d}{2} \) is divided into two equal parts, each of length \( \frac{d}{4} \). The first part is covered at a speed of \( 4 \, \text{m/s} \) and the second part at a speed of \( 3 \, \text{m/s} \).

### Step 4: Calculate time for the second half
1. For the first quarter of the second half:
\[ t_2 = \frac{\frac{d}{4}}{4} = \frac{d}{16} \]

2. For the second quarter of the second half:
\[ t_3 = \frac{\frac{d}{4}}{3} = \frac{d}{12} \]

### Step 5: Calculate total time
Now, we can find the total time taken to cover the entire distance \( d \):
\[ \text{Total Time} = t_1 + t_2 + t_3 = \frac{d}{10} + \frac{d}{16} + \frac{d}{12} \]

### Step 6: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of \( 10, 16, \) and \( 12 \) is \( 240 \).

1. Convert each fraction:
- \( \frac{d}{10} = \frac{24d}{240} \)
- \( \frac{d}{16} = \frac{15d}{240} \)
- \( \frac{d}{12} = \frac{20d}{240} \)

2. Now add them:
\[ \text{Total Time} = \frac{24d}{240} + \frac{15d}{240} + \frac{20d}{240} = \frac{(24 + 15 + 20)d}{240} = \frac{59d}{240} \]

### Step 7: Calculate average velocity
The average velocity \( v_{avg} \) is given by the formula:
\[ v_{avg} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{d}{\frac{59d}{240}} = \frac{240}{59} \approx 4.07 \, \text{m/s} \]

### Final Answer
The average velocity of the body is approximately \( 4.07 \, \text{m/s} \).

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