Two planes move along a circle of circumference 1.2 kms with constant speeds. When they move in different directions, they meet every 15 seconds and when they move in the same direction one plane overtakes the other every 60 seconds. The speed of the slower plane is

### Concept & Relative Speed on a Circle When two bodies move along a circular path, their relative speed depends on their direction. If moving in opposite directions, their relative speed is the sum of their speeds. If moving in the same direction, it is the difference. The distance covered to meet or overtake is the circumference of the circle. $$Distance = Relative\ Speed \times Time$$ ### Step-by-Step Solution 1. Let the speed of the faster plane be $u$ km/s and the slower plane be $v$ km/s. 2. **Opposite directions:** They meet every 15 seconds. Distance = $1.2$ km. Relative speed = $(u + v)$ $$(u + v) = \frac{1.2}{15} = 0.08 \text{ km/s}$$ 3. **Same direction:** The faster plane overtakes the slower one every 60 seconds. Relative speed = $(u - v)$ $$(u - v) = \frac{1.2}{60} = 0.02 \text{ km/s}$$ 4. **Solve for v:** We need the speed of the slower plane. Subtract the second equation from the first. $$(u + v) - (u - v) = 0.08 - 0.02$$ $$2v = 0.06$$ $$v = 0.03 \text{ km/s}$$ ### Exam Strategy & Shortcut Once you establish the relative speeds ($0.08$ for sum, $0.02$ for difference), simply subtract the difference from the sum and divide by 2 to find the smaller value directly: $\frac{0.08 - 0.02}{2} = 0.03$. ### Common Pitfall Mixing up units. Ensure you notice that circumference is in "kms" and time is in "seconds", aligning perfectly with the options in "km/s". Do not needlessly convert to km/hr. ### Final Answer Therefore, the correct answer is **0.03 km/s**.