Circles
Motion on Circles
Grade 11
Question:
<p>A circle of radius unity is centred at origin. Two particles start moving at the same time from the point \((1, 0)\) and move around the circle in opposite direction. One of the particle moves counter clockwise with constant speed u and the other moves clockwise with constant speed 3u. After leaving \((1, 0)\), the two particles meet first at a point P, and continue until they meet next at point Q. The coordinates of the point Q are:</p>
<p>(a) \((1, 0)\)</p>
<p>(b) \((0, 1)\)</p>
<p>(c) \((0, -1)\)</p>
<p>(d) \((-1, 0)\)</p>
Step-by-Step Solution
Key Concept: Use relative speed and arc length on the unit circle to find meeting points.
<p>The relative speed of approach is \(u + 3u = 4u\). The circumference of the unit circle is \(2\pi\). Time to first meeting: \(t_1 = \frac{2\pi}{4u} = \frac{\pi}{2u}\). In this time, the counter-clockwise particle travels arc length \(u \cdot \frac{\pi}{2u} = \frac{\pi}{2}\) radians, reaching point \((0,1)\) which is point P. For the second meeting, they must be together again. The total distance covered by both together must equal \(2\pi\) again. Time to second meeting: \(t_2 = 2t_1 = \frac{\pi}{u}\). The counter-clockwise particle travels \(\pi\) radians from \((1,0)\), reaching \((-1,0)\), which is point Q.</p>
Correct Answer: D