A CAR ON THE CROSS ROAD IS MOVING TOWARDS EAST AT A SPEED OF 40 km/hr. ANOTHER CAR WHICH IS 1 km SOUTH OF THE CROSSING IS APPROACHING THE CROSSING WITH THE SAME SPEED. THE CLOSEST DISTANCE OF APPROACH IS WHAT ?

To solve the problem of finding the closest distance of approach between the two cars, we can visualize their movements on a coordinate system. Let's break down the scenario step-by-step.

Understanding the Positions

We have two cars: Car A is moving east towards the crossing at a speed of 40 km/hr, and Car B is positioned 1 km south of the crossing, also moving towards the crossing at the same speed.

Setting Up the Coordinate System

Let's define the crossing point as the origin (0,0) on a Cartesian coordinate system:

• Car A starts at (0, 0) and moves east along the x-axis.
• Car B starts at (0, -1) (1 km south of the crossing) and moves north along the y-axis.

Equations of Motion

We can express the positions of the cars as functions of time (t), measured in hours:

• Position of Car A: (40t, 0)
• Position of Car B: (0, -1 + 40t)

Finding the Distance Between the Cars

The distance (D) between the two cars at any time can be found using the distance formula:

D = √[(x2 - x1)² + (y2 - y1)²]

Substituting the positions of the two cars, we have:

D = √[(40t - 0)² + (0 - (-1 + 40t))²]

This simplifies to:

D = √[(40t)² + (1 - 40t)²]

Expanding the Distance Formula

Now, let's expand the squared terms:

• (40t)² = 1600t²
• (1 - 40t)² = 1 - 80t + 1600t²

Combining these gives us:

D² = 1600t² + 1 - 80t + 1600t² = 3200t² - 80t + 1

Minimizing the Distance

To find the closest distance of approach, we need to minimize D². This can be done using calculus. We will take the derivative of D² with respect to t, set it to zero, and solve for t:

d(D²)/dt = 6400t - 80 = 0

Simplifying this gives:

6400t = 80

t = 80 / 6400 = 0.0125 hours (or 0.75 minutes)

Calculating the Minimum Distance

Now, we can substitute t back into the distance formula to find the minimum distance:

D² = 3200(0.0125)² - 80(0.0125) + 1

D² = 3200(0.00015625) - 1 + 1 = 0.5

Hence, D = √0.5 = 0.707 km (approximately 707 meters).

Final Result

The closest distance of approach between the two cars is approximately 0.707 km, or 707 meters. This distance reflects the minimum separation they achieve while moving towards the crossing. Understanding these relationships and calculations can help in various real-life applications, such as traffic management and collision avoidance systems.