A car moves a distance of `200 m`. It covers the first-half of the distance at speed `40km//h` and the second-half of distance at speed `v km//h`. The average speed is `48km//h`. Find the value of v.

Options

A. `56km//h`
B. `60 km//h`
C. `50km//h`
D. `48 km//h`

Answer

To solve the problem, we need to find the speed \( v \) at which the car travels the second half of the distance, given that the average speed for the entire journey is \( 48 \text{ km/h} \).

### Step-by-Step Solution:

1. **Identify the total distance and split it into halves**:
The total distance covered by the car is \( 200 \text{ m} \). Therefore, the first half is:
\[ d_1 = \frac{200}{2} = 100 \text{ m} \]
and the second half is:
\[ d_2 = \frac{200}{2} = 100 \text{ m} \]

2. **Convert speeds from km/h to m/s**:
The speed of the car for the first half is given as \( 40 \text{ km/h} \). To convert this to meters per second:
\[ 40 \text{ km/h} = \frac{40 \times 1000}{3600} = \frac{40000}{3600} \approx 11.11 \text{ m/s} \]
The average speed is \( 48 \text{ km/h} \):
\[ 48 \text{ km/h} = \frac{48 \times 1000}{3600} = \frac{48000}{3600} \approx 13.33 \text{ m/s} \]

3. **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:
\[ t_1 = \frac{d_1}{\text{speed}_1} = \frac{100 \text{ m}}{11.11 \text{ m/s}} \approx 9 \text{ seconds} \]

4. **Let the speed for the second half be \( v \) in m/s**:
The time taken for the second half of the distance is:
\[ t_2 = \frac{d_2}{v} = \frac{100 \text{ m}}{v} \]

5. **Calculate the total time taken**:
The total time taken for the journey is:
\[ t_{total} = t_1 + t_2 = 9 + \frac{100}{v} \]

6. **Use the average speed formula**:
The average speed is given by the total distance divided by the total time:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{200}{t_{total}} = 48 \text{ km/h} \]
Converting \( 48 \text{ km/h} \) to m/s gives us:
\[ 48 \text{ km/h} = 13.33 \text{ m/s} \]

7. **Set up the equation**:
Now, substituting the values into the average speed equation:
\[ 13.33 = \frac{200}{9 + \frac{100}{v}} \]

8. **Cross-multiply and solve for \( v \)**:
Rearranging gives:
\[ 13.33 \left(9 + \frac{100}{v}\right) = 200 \]
Expanding this:
\[ 120 + \frac{1333}{v} = 200 \]
Simplifying:
\[ \frac{1333}{v} = 80 \]
Thus:
\[ v = \frac{1333}{80} \approx 16.66 \text{ m/s} \]

9. **Convert \( v \) back to km/h**:
To find \( v \) in km/h:
\[ v \approx 16.66 \text{ m/s} \times \frac{3600}{1000} \approx 60 \text{ km/h} \]

### Final Answer:
The value of \( v \) is approximately \( 60 \text{ km/h} \).