Final answer:
The minimum speed of a roller coaster at the top of a 15-meter high loop to maintain motion without falling is calculated using the radius and acceleration due to gravity, resulting in approximately 12.13 meters per second.
Step-by-step explanation:
Calculating the Speed of a Roller Coaster at the Top of a Loop
When considering the roller coaster's motion at the top of a loop, physics principles such as centripetal force and potential energy are important. To calculate the roller coaster speed at the top of the loop, we need to consider that the total acceleration experienced by the passengers is the vector sum of the gravitational acceleration (1 g) and the centripetal acceleration due to the motion of the coaster (1.50 g). Since we want to find the minimum speed required at the top of the loop, we only need to consider the acceleration due to gravity being 1 g for this calculation.
The formula to find the minimum speed (v) at the top of the loop using the radius of curvature (r) and the acceleration due to gravity (g) is:
v = √(rg)
Given the radius of curvature (15.0 m) and the acceleration due to gravity (9.8 m/s2), the minimum speed at the top of the loop can be calculated as follows:
v = √(15.0 m × 9.8 m/s2)
v = √(147 m2/s2)
v = 12.13 m/s
Hence, the minimum speed of the roller coaster at the top of the 15-meter high loop to maintain the motion without falling due to gravity is approximately 12.13 meters per second.