A roller coaster is designed such that riders experience "weightlessness" as they go round the top of a hill whose radius of curvature is `20m`. The speed of the car at the top of the hill is between

Options

A. `14m//s` and `15m//s`
B. `15m//s` and `16m//s`
C. `16m//s` and `17m//s`
D. `13m//s` and `14m//s`

Explanation

To determine the speed of the roller coaster car at the top of the hill where riders experience "weightlessness," we can use the concept of centripetal force and gravitational force.

### Step-by-Step Solution:

1. **Understanding Weightlessness**:
- A rider experiences weightlessness when the normal force acting on them is zero. At the top of the hill, the only force acting on the rider is the gravitational force, which provides the necessary centripetal force to keep the rider moving in a circular path.

2. **Forces Acting at the Top of the Hill**:
- At the top of the hill, the gravitational force (mg) must equal the centripetal force required to keep the rider in circular motion.
- The centripetal force is given by the formula:
\[ F_c = \frac{mv^2}{r} \]
- Where:
- \( m \) = mass of the rider
- \( v \) = speed of the roller coaster
- \( r \) = radius of curvature (20 m in this case)

3. **Setting Up the Equation**:
- At the point of weightlessness, the gravitational force is equal to the centripetal force:
\[ mg = \frac{mv^2}{r} \]
- We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ g = \frac{v^2}{r} \]

4. **Solving for Speed (v)**:
- Rearranging the equation gives us:
\[ v^2 = gr \]
- Taking the square root of both sides, we find:
\[ v = \sqrt{gr} \]

5. **Substituting Known Values**:
- We know \( g \) (acceleration due to gravity) is approximately \( 9.8 \, \text{m/s}^2 \) and \( r = 20 \, \text{m} \):
\[ v = \sqrt{9.8 \times 20} \]
\[ v = \sqrt{196} \]
\[ v \approx 14 \, \text{m/s} \]

6. **Determining the Range of Speed**:
- To ensure weightlessness, the speed must be less than or equal to this value. Therefore, the speed of the roller coaster at the top of the hill must be less than or equal to approximately \( 14 \, \text{m/s} \).

### Final Answer:
The speed of the car at the top of the hill is between \( 0 \, \text{m/s} \) and \( 14 \, \text{m/s} \).