Circular Track Length Calculation

This solution determines the length of a circular track by analyzing the relative motion of two runners.

Given Information

• Runner A's speed: $v_A = 8\text{ km/h}$
• Runner B's speed: $v_B = 6\text{ km/h}$
• Catch-up time: $t = 30\text{ minutes}$
• Starting positions: Diametrically opposite points
• Direction of running: Same

Calculations

We calculate the track length using the concept of relative speed. Since both runners move in the same direction, runner A gains on runner B at a speed equal to the difference between their speeds.

• Relative Speed Calculation: The relative speed ($v_{rel}$) of runner A with respect to runner B is: $ v_{rel} = v_A - v_B $ $ v_{rel} = 8\text{ km/h} - 6\text{ km/h} = 2\text{ km/h} $
• Time Conversion: The time taken for A to catch B is given in minutes. Convert this to hours: $ t = 30\text{ minutes} = \frac{30}{60}\text{ hours} = 0.5\text{ hours} $
• Relative Distance Covered: The relative distance covered by runner A to catch runner B is the product of their relative speed and the time taken: $ \text{Relative Distance} = v_{rel} \times t $ $ \text{Relative Distance} = 2\text{ km/h} \times 0.5\text{ h} = 1\text{ km} $
• Track Length Determination: Since the runners start at diametrically opposite points, the initial separation is half the track's circumference ($L/2$). When runner A catches up with runner B for the first time, the relative distance covered must equal this initial separation. $ \text{Relative Distance} = \frac{L}{2} $ $ 1\text{ km} = \frac{L}{2} $ Solving for the track length ($L$): $ L = 2 \times 1\text{ km} $ $ L = 2\text{ km} $

Result

The length of the circular track is $2\text{ km}$.