Final answer:
The greatest speed at which the stuntman can drive without the car leaving the road at the top of a circular hill is calculated using the centripetal force formula. It involves setting the gravitational force equal to the centripetal force and solving for velocity. The maximum speed for the given radius of 250 meters is approximately 49.5 m/s.
Step-by-step explanation:
The student is asking about a scenario involving circular motion and the physics concept of centripetal force. When a stuntman drives a car over the top of a hill that is shaped like a circular arc, the car will experience a centripetal force directed towards the center of the circle, which in this case is due to gravity pulling the car towards the ground. The maximum speed at which the car can travel without leaving the ground is when the centripetal force is equal to the gravitational force acting on the car at the top of the hill.
To calculate this maximum speed, we use the formula for centripetal force Fc = m * v2 / r, where m is the mass of the car, v is the velocity, and r is the radius of the circular arc. At the top of the hill, the only 'downward' force acting on the car is its weight, which is equal to the gravitational force mg, where g is the acceleration due to gravity (9.81 m/s2). Setting the centripetal force equal to the gravitational force gives us m * v2 / r = m * g, which simplifies to v2 = r * g.
Substituting the given radius of 250 m into the simplified equation, we get v2 = 250 * 9.81. Taking the square root of both sides, we find the maximum speed v that the car can travel without leaving the road is approximately 49.5 m/s.