Calculating the Distance and Time for a Hoop Rolling Up an Inclined Plane

Explanation:

The hoop will rise approximately 0.57 meters up the incline and will take approximately 0.26 seconds to arrive back at the bottom. The hoop will roll up the incline to a certain distance. To determine this distance, we need to consider the conservation of mechanical energy. The initial kinetic energy of the hoop is converted into potential energy as it moves up the incline. We can use the equation: mgh = 0.5mv² where m is the mass of the hoop, g is the acceleration due to gravity, h is the vertical distance the hoop rises, and v is the initial velocity of the hoop. Solving for h, we get: h = (v²)/(2g) Substituting the given values, we have: h = (3.3²)/(2*9.8) Calculating this, we find that the hoop will rise approximately 0.57 meters up the incline. To determine the time it takes for the hoop to arrive back at the bottom of the incline, we can use the formula: t = sqrt((2h)/(g*sin(θ))) where t is the time, h is the vertical distance traveled, g is the acceleration due to gravity, and θ is the angle of the incline. Substituting the values, we get: t = sqrt((2*0.57)/(9.8*sin(15))) Calculating this, we find that the hoop will take approximately 0.26 seconds to arrive back at the bottom.