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 solve the problem, we need to calculate the average velocity of the body that moves in a straight line, covering different distances at different speeds. Here’s a step-by-step solution:

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

### Step 2: Calculate time for the first half
Using the formula for time \( t = \frac{d}{v} \):
- Distance for the first half = \( \frac{s}{2} \)
- Speed for the first half = \( 5 \, \text{m/s} \)

\[ t_1 = \frac{\frac{s}{2}}{5} = \frac{s}{10} \]

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

### Step 4: Calculate time for the second half
For the first part of the second half (from A to B):
- Speed = \( 4 \, \text{m/s} \)

\[ t_2 = \frac{\frac{s}{4}}{4} = \frac{s}{16} \]

For the second part of the second half (from B to C):
- Speed = \( 3 \, \text{m/s} \)

\[ t_3 = \frac{\frac{s}{4}}{3} = \frac{s}{12} \]

### Step 5: Calculate total time
Now, we can find the total time taken for the entire journey:

\[ T = t_1 + t_2 + t_3 = \frac{s}{10} + \frac{s}{16} + \frac{s}{12} \]

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

Converting each term:
- \( \frac{s}{10} = \frac{24s}{240} \)
- \( \frac{s}{16} = \frac{15s}{240} \)
- \( \frac{s}{12} = \frac{20s}{240} \)

Now, we can add them:

\[ T = \frac{24s}{240} + \frac{15s}{240} + \frac{20s}{240} = \frac{(24 + 15 + 20)s}{240} = \frac{59s}{240} \]

### Step 7: Calculate average velocity
The average velocity \( V_{avg} \) is given by the formula:

\[ V_{avg} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{s}{T} \]

Substituting the value of \( T \):

\[ V_{avg} = \frac{s}{\frac{59s}{240}} = \frac{240}{59} \]

### Step 8: Calculate the numerical value
Now, we can calculate \( \frac{240}{59} \):

\[ \frac{240}{59} \approx 4.07 \, \text{m/s} \]

### Conclusion
Thus, the average velocity of the body is nearly equal to \( 4 \, \text{m/s} \).

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