Pathfinder for Olympiad and JEE Advanced Physics
LEVELJEE Advanced
Visualized Solution
The Sigma Insight: Motion in a Plane
The problem of two bikers racing on a triangular track is a beautiful blend of basic kinematics and number theory. It tests your ability to break down a physical situation into mathematical constraints. Let's dive into the step-by-step logical breakdown of this fascinating problem.
Analyzing the Setup
Imagine you are standing at point A of the triangular track ABC. Two bikers start their engines and take off in opposite directions. The first biker takes the clockwise route, heading straight down the hypotenuse towards point B. The second biker takes the anticlockwise route, navigating through the right-angled corner at C before finally reaching B.
We are given that both bikers arrive at point B at the exact same instant, exactly after they started. This is our crucial first clue. To understand their future motion, we first need to determine how fast each biker is traveling.
The clockwise biker travels along the hypotenuse, which has a length of . Therefore, their speed is simply the distance divided by the time:
The anticlockwise biker, on the other hand, has to cover the two legs of the triangle. The total distance they cover is . Their speed is:
The Master Equation
Now that we have their speeds, the race enters its second phase. Both bikers are now at point B, and they continue riding around the track. The question asks when they will simultaneously cross point B again.
For a biker to return to point B, they must complete a full lap of the track. The total perimeter of the triangular track is .
Let's calculate the time it takes for each biker to complete one single lap.
For the first biker:
For the second biker:
Final Calculation
Here is where the magic happens. For both bikers to be at point B at the exact same time, the elapsed time must be a perfect multiple of both their individual lap times. In mathematics, the smallest number that is a multiple of two other numbers is their Least Common Multiple (LCM).
So, the time until their next meeting at B is:
Finding the LCM of fractions might seem tricky, but there is a simple rule: take the LCM of the numerators and divide it by the Highest Common Factor (HCF) of the denominators.
The LCM of 48 and 48 is simply 48. The HCF of 5 and 7 (which are prime to each other) is 1.
And there we have it! Exactly 48 minutes after their first meeting, the two bikers will cross point B together once again. This problem elegantly shows how physical periodic motion perfectly maps onto the mathematical concept of LCM.
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