Two highways are perpendicular to each other imagine them to be along the x-axis and the y-axis, respectively. At the instant t = 0, a police car P is at a distance d = 400 m from the intersection and moving at speed of 8 km//h towards it at a speed of 60 km//h along they y-axis. The minimum distance between the cars is

Options

A. `300 m`
B. `240 m`
C. `180 m`
D. `120 m`

Explanation

To solve the problem of finding the minimum distance between the two cars, we can follow these steps:

### Step 1: Understand the scenario
We have two cars:
- Car A (the police car) is moving towards the intersection along the y-axis from a distance of 400 m at a speed of 80 km/h.
- Car B is moving along the x-axis at a speed of 60 km/h.

### Step 2: Convert speeds to consistent units
Convert the speeds from km/h to m/s for easier calculations:
- Speed of Car A (VA) = 80 km/h = (80 * 1000 m) / (3600 s) = 22.22 m/s
- Speed of Car B (VB) = 60 km/h = (60 * 1000 m) / (3600 s) = 16.67 m/s

### Step 3: Determine the relative velocity
To find the minimum distance, we need to analyze the relative motion of the two cars. The relative velocity of Car A with respect to Car B can be represented as:
- \( V_{AB} = V_A - V_B \)

Since Car A is moving towards the intersection (in the negative y-direction) and Car B is moving towards the intersection (in the positive x-direction), we can consider:
- \( V_A = -22.22 \hat{j} \) (moving downwards)
- \( V_B = 16.67 \hat{i} \) (moving right)

### Step 4: Calculate the angle of approach
The angle \( \theta \) between the paths of the two cars can be determined using the tangent function:
- \( \tan(\theta) = \frac{V_B}{V_A} = \frac{16.67}{22.22} \)

### Step 5: Find the sine of the angle
Using the Pythagorean theorem, we can find the hypotenuse:
- \( \text{Hypotenuse} = \sqrt{(V_A^2 + V_B^2)} = \sqrt{(22.22^2 + 16.67^2)} \)

Now calculate:
- \( V_A^2 = 493.83 \)
- \( V_B^2 = 278.89 \)
- \( \text{Hypotenuse} = \sqrt{(493.83 + 278.89)} = \sqrt{772.72} \approx 27.8 \)

Now, we can find \( \sin(\theta) \):
- \( \sin(\theta) = \frac{V_B}{\text{Hypotenuse}} = \frac{16.67}{27.8} \)

### Step 6: Calculate the minimum distance
The minimum distance \( d_{min} \) can be calculated using the formula:
- \( d_{min} = d \cdot \sin(\theta) \)
Where \( d = 400 \) m.

Now substituting the values:
- \( d_{min} = 400 \cdot \sin(\theta) \)

### Step 7: Final calculation
Using the value of \( \sin(\theta) \) calculated above, we can find \( d_{min} \).