TeXes Science Practice Test

How fast must a roller coaster car navigate a loop of 20m height to ensure a moment of weightlessness?

1. 5m/s

2. 10m/s

3. 15m/s

4. 20m/s

The correct answer is: 10m/s

To experience a moment of weightlessness while navigating a vertical loop, the roller coaster car must maintain a certain minimum speed at the top of the loop. This is because, at the very top of the loop, the gravitational force acting on the roller coaster must be countered by the required centripetal force to keep the car moving along the circular path of the loop. The necessary condition for weightlessness is that the force due to gravity is equal to the centripetal force. At the top of the loop, the gravitational force can be expressed as \( mg \), where \( m \) is the mass of the roller coaster car, and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)). The centripetal force required to keep the car on the circular path is given by \( \frac{mv^2}{r} \), where \( v \) is the speed of the car and \( r \) is the radius of the loop. Given that the height of the loop is 20 m, the radius \( r \) can be determined as half of the height, so \( r = 10 \, \text{m}