To tackle the problem involving Ram and Shyam walking on perpendicular tracks, we first need to analyze their movements and the distances involved. Let's break it down step by step to find the shortest distance between them and the time at which this occurs.
Understanding Their Movements
Ram is walking towards the intersection at a speed of 3 m/s, starting from a distance of 20 m. Shyam is moving towards the same intersection at a speed of 4 m/s, starting from 40 m away. Since they are on perpendicular tracks, we can visualize their paths as forming a right triangle, where the intersection is the right angle.
Position Equations
We can express their positions as functions of time:
- Position of Ram at time t: x_R(t) = 20 - 3t
- Position of Shyam at time t: y_S(t) = 40 - 4t
Finding the Shortest Distance
The distance between Ram and Shyam at any time t can be represented using the Pythagorean theorem:
D(t) = √((x_R(t))² + (y_S(t))²)
Substituting the position equations into the distance formula gives us:
D(t) = √((20 - 3t)² + (40 - 4t)²)
Expanding the Distance Formula
Let's expand this expression:
- D(t) = √((20 - 3t)² + (40 - 4t)²)
- = √((400 - 120t + 9t²) + (1600 - 320t + 16t²))
- = √(400 + 1600 - 120t - 320t + 25t²)
- = √(2000 - 440t + 25t²)
Minimizing the Distance
To find the shortest distance, we need to minimize D(t). This can be done by minimizing the square of the distance, D²(t), which simplifies our calculations:
D²(t) = 2000 - 440t + 25t²
To find the minimum, we can take the derivative and set it to zero:
d(D²)/dt = -440 + 50t = 0
Solving for t gives:
50t = 440 → t = 8.8 seconds
Calculating the Shortest Distance
Now, substituting t = 8.8 seconds back into the distance formula:
D(8.8) = √(2000 - 440(8.8) + 25(8.8)²)
Calculating this step-by-step:
- D(8.8) = √(2000 - 3872 + 1936)
- = √(64) = 8 m
Final Answers
From our calculations, we find:
- For Q2, the shortest distance between Ram and Shyam is 8 m, which corresponds to option D.
- For Q3, the time at which they are at the shortest distance is 8.8 seconds, which corresponds to option A.
This systematic approach allows us to derive both the shortest distance and the time at which it occurs by using their speed and initial positions effectively. If you have any further questions or need clarification on any part of the process, feel free to ask!
