Final answer:
The minimum speed a roller coaster can have without falling from the track while upside down is 17.15 m/s for a loop with a 30 m radius of curvature, calculated using principles of circular motion and centripetal force where the only force at the top of the loop is gravity.
Step-by-step explanation:
The minimum speed a roller coaster car can travel without falling from the track when it is upside down relates to the concepts of circular motion and centripetal force. We need to consider physics principles, specifically the relationship between centripetal force, gravitational force, and the mass and velocity of the roller coaster car. At the minimum speed, when the roller coaster is at the top of a loop, the only force acting as the centripetal force is gravity itself. The minimum speed can be calculated using the formula for centripetal acceleration ac = v2/r, where v is the velocity and r is the radius of curvature. The centripetal acceleration at the top of the loop must be equal to the acceleration due to gravity g (9.81 m/s2), hence v2/r = g. Solving for v gives us the minimum velocity v = sqrt(g * r). For a radius of curvature of 30 m, the minimum speed v = sqrt(9.81 m/s2 * 30 m) = sqrt(294.3 m2/s2) = 17.15 m/s. Therefore, the minimum speed is approximately 17.15 meters per second to ensure the coaster car does not fall off the track.