A car moves a distance of 200m. It covers first half of the distance at speed `60kmh^(-1)` and the second half at speed v. If the average speed is `40kmh^(-1)`, the value of v is

Options

A. `30kmh^(-1)`
B. `13 kmh^(-1)`
C. `60kmh^(-1)`
D. `40kmh^(-1)`

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Problem
The car travels a total distance of 200 meters. It covers the first half (100 meters) at a speed of 60 km/h and the second half (100 meters) at an unknown speed \( v \). The average speed of the entire journey is given as 40 km/h.

### Step 2: Convert Units
Since the distance is given in meters, we should convert the speeds from km/h to m/s for consistency.
- \( 60 \text{ km/h} = \frac{60 \times 1000}{3600} = 16.67 \text{ m/s} \)
- \( 40 \text{ km/h} = \frac{40 \times 1000}{3600} = 11.11 \text{ m/s} \)

### Step 3: Calculate Total Distance and Average Speed
The total distance \( D \) is 200 meters. The average speed \( V_{avg} \) is given as 40 km/h (or 11.11 m/s).

### Step 4: Use the Formula for Average Speed
The formula for average speed is:
\[ V_{avg} = \frac{D}{T_{total}} \]
Where \( T_{total} \) is the total time taken for the journey. Rearranging gives:
\[ T_{total} = \frac{D}{V_{avg}} = \frac{200 \text{ m}}{11.11 \text{ m/s}} \approx 18 \text{ seconds} \]

### Step 5: Calculate Time for Each Half of the Journey
1. **Time for the first half (100 m at 60 km/h):**
\[ T_1 = \frac{100 \text{ m}}{16.67 \text{ m/s}} \approx 6 \text{ seconds} \]

2. **Time for the second half (100 m at speed \( v \)):**
\[ T_2 = \frac{100 \text{ m}}{v \text{ m/s}} \]

### Step 6: Set Up the Equation for Total Time
The total time for the journey can be expressed as:
\[ T_{total} = T_1 + T_2 \]
Substituting the known values:
\[ 18 = 6 + \frac{100}{v} \]

### Step 7: Solve for \( v \)
Rearranging the equation gives:
\[ \frac{100}{v} = 18 - 6 = 12 \]
Now, solving for \( v \):
\[ v = \frac{100}{12} \approx 8.33 \text{ m/s} \]

### Step 8: Convert \( v \) Back to km/h
To convert \( v \) back to km/h:
\[ v = 8.33 \text{ m/s} \times \frac{3600}{1000} \approx 30 \text{ km/h} \]

### Final Answer
The value of \( v \) is **30 km/h**.

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