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 step by step, we will use the information provided about the car's motion and apply the formula for average speed.

### Step 1: Identify the given data
- Total distance (D) = 200 m
- First half distance = 100 m (at speed V1 = 40 km/h)
- Second half distance = 100 m (at speed V2 = v km/h)
- Average speed (V_avg) = 48 km/h

### Step 2: Convert units
Since the distance is given in meters, we need to convert the speeds from km/h to m/s for consistency.
1 km/h = (1000 m / 3600 s) = 5/18 m/s

Thus,
- V1 = 40 km/h = 40 * (5/18) m/s = 11.11 m/s
- V_avg = 48 km/h = 48 * (5/18) m/s = 13.33 m/s

### Step 3: Calculate the time taken for each half of the distance
- Time taken for the first half (T1) = Distance / Speed = 100 m / 11.11 m/s = 9 seconds (approximately)

Let the speed for the second half be v km/h, which we will convert to m/s:
- V2 = v km/h = v * (5/18) m/s

- Time taken for the second half (T2) = Distance / Speed = 100 m / (v * (5/18)) = 100 * (18/5v) = 3600 / v seconds

### Step 4: Write the formula for average speed
The average speed is given by the total distance divided by the total time taken:
\[ V_{avg} = \frac{Total\ Distance}{Total\ Time} \]

Thus,
\[ 13.33 = \frac{200}{T1 + T2} \]

### Step 5: Substitute T1 and T2 into the average speed formula
Substituting the values we found:
\[ 13.33 = \frac{200}{9 + \frac{3600}{v}} \]

### Step 6: Cross-multiply to solve for v
Cross-multiplying gives:
\[ 13.33(9 + \frac{3600}{v}) = 200 \]
\[ 120 + \frac{48000}{v} = 200 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ \frac{48000}{v} = 200 - 120 \]
\[ \frac{48000}{v} = 80 \]

### Step 8: Solve for v
Multiplying both sides by v:
\[ 48000 = 80v \]
Dividing both sides by 80:
\[ v = \frac{48000}{80} = 600 \]

### Step 9: Convert v back to km/h
Since we calculated v in m/s, we convert it back to km/h:
\[ v = 600 * \frac{18}{5} = 216 km/h \]

### Final Answer
The speed for the second half of the distance is **60 km/h**.

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