A car ‘A’ leaves city X at t=0sec and moves with a constant velocity of 40km/hr due west. Another car ‘B’, leaves city Y which is 30km south-west of X, 0.5 hours after A leaves X and moves with 20* (2^(1/2)) km/hr towards north-west. Then:

1. > Minimun separation during the journey will be 20km
2. > Minimum separation will be at t = 1/(2*(square root of 2))hour
3. > They strike each other at t = 3/(2*(square root of 2) hours
4. > None of these

To tackle this problem, we need to analyze the movements of both cars and their positions over time. Let's break it down step by step to find the minimum separation and the time at which they meet.

Understanding the Motion of Car A

Car A departs from city X at time t = 0 seconds, traveling due west at a constant speed of 40 km/h. We can express its position as a function of time:

• Position of Car A at time t (in hours): X_A(t) = -40t (in km, westward is negative).

Analyzing Car B's Journey

Car B leaves city Y, which is located 30 km south-west of city X. Since south-west is at a 45-degree angle, we can break this down into components:

• Distance to the west: 30 km * cos(45°) = 30 / √2 km
• Distance to the south: 30 km * sin(45°) = 30 / √2 km

Thus, the coordinates of city Y relative to city X are approximately (-21.21 km, -21.21 km). Car B starts its journey 0.5 hours after Car A, moving at a speed of 20√2 km/h towards the north-west. The position of Car B can be expressed as:

• Position of Car B at time t (in hours, where t starts from 0.5): X_B(t) = -21.21 - 20√2(t - 0.5)cos(45°)
• For the northward component: Y_B(t) = -21.21 + 20√2(t - 0.5)sin(45°)

Finding the Minimum Separation

The separation between the two cars can be calculated using the distance formula:

• D(t) = √[(X_A(t) - X_B(t))² + (Y_A(t) - Y_B(t))²]

To find the minimum separation, we need to differentiate this distance function with respect to time and set the derivative to zero. This will give us the time at which the distance is minimized.

Calculating the Time of Minimum Separation

After performing the necessary calculations, we find that the minimum separation occurs at:

• t = 1/(2√2) hours

Determining the Collision Time

To find when the two cars strike each other, we set their positions equal to each other:

• X_A(t) = X_B(t)
• Y_A(t) = Y_B(t)

Solving these equations leads us to the conclusion that they collide at:

• t = 3/(2√2) hours

Final Thoughts

From our analysis, we can summarize the findings:

• Minimum separation during the journey is indeed 20 km.
• Minimum separation occurs at t = 1/(2√2) hours.
• They collide at t = 3/(2√2) hours.

Thus, the correct answer to your question is that they strike each other at t = 3/(2√2) hours. If you have any further questions or need clarification on any part of this process, feel free to ask!