A body covers half the distance with a velocity 10 m/ s and remaining half with a velocity 15 m/s along a straight line. The average velocity will be-

Options

A. 12 m/s
B. 10 m/s
C. 5 m/s
D. 12.5 m/s

Answer

To solve the problem of finding the average velocity of a body that covers half the distance with a velocity of 10 m/s and the remaining half with a velocity of 15 m/s, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Distance**:
Let the total distance be \( d \). Therefore, half of the distance is \( \frac{d}{2} \).

2. **Calculate Time for the First Half**:
The body covers the first half of the distance (\( \frac{d}{2} \)) with a velocity of 10 m/s.
The time taken to cover this distance can be calculated using the formula:
\[ t_1 = \frac{\text{distance}}{\text{velocity}} = \frac{\frac{d}{2}}{10} = \frac{d}{20} \]

3. **Calculate Time for the Second Half**:
The body covers the second half of the distance (\( \frac{d}{2} \)) with a velocity of 15 m/s.
The time taken to cover this distance is:
\[ t_2 = \frac{\text{distance}}{\text{velocity}} = \frac{\frac{d}{2}}{15} = \frac{d}{30} \]

4. **Calculate Total Time**:
The total time taken to cover the entire distance is the sum of the times for both halves:
\[ t_{\text{total}} = t_1 + t_2 = \frac{d}{20} + \frac{d}{30} \]

5. **Find a Common Denominator**:
To add the fractions, we need a common denominator. The least common multiple of 20 and 30 is 60.
\[ t_{\text{total}} = \frac{3d}{60} + \frac{2d}{60} = \frac{5d}{60} = \frac{d}{12} \]

6. **Calculate Average Velocity**:
The average velocity (\( v_{\text{avg}} \)) is defined as the total distance divided by the total time:
\[ v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{d}{\frac{d}{12}} = 12 \text{ m/s} \]

### Final Answer:
The average velocity of the body is **12 m/s**.

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