A roller coaster is designed such that riders experience "weightlessness" as they go around the top of a hill

Question

A roller coaster is designed such that riders experience "weightlessness" as they go around the top of a hill whose radius of curvature is $20 	ext{ m}$. The speed of the car at the top of the hill is between:

Accepted Answer

1. **Condition for Weightlessness:** Riders experience weightlessness at the top of the hill when the normal reaction force ($N$) exerted by the seat on them becomes zero. In this condition, the gravitational force alone provides the necessary centripetal force required for circular motion.
2. **Equation of Motion:** At the top of the vertical path:
   $$ mg - N = \frac{mv^2}{R} $$
   Setting $N = 0$ for weightlessness:
   $$ mg = \frac{mv^2}{R} $$
   $$ v = \sqrt{Rg} $$
3. **Calculation:** Given $R = 20 	ext{ m}$ and approximating $g \approx 9.8 	ext{ m/s}^2$ to $10 	ext{ m/s}^2$:
   $$ v = \sqrt{20 	imes 10} = \sqrt{200} \approx 14.14 	ext{ m/s} $$
   Using $g = 9.8 	ext{ m/s}^2$:
   $$ v = \sqrt{20 	imes 9.8} = \sqrt{196} = 14 	ext{ m/s} $$
4. **Conclusion:** The speed is approximately $14.14 	ext{ m/s}$, which falls between $14 	ext{ m/s}$ and $15 	ext{ m/s}$.
   (Reference: NCERT Class 11, Physics Part I, Chapter 5: Laws of Motion, Section 5.10 Circular Motion).

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