Question

A solid ball rolls smoothly without slipping from rest starting at height h = 22.8 m until it leaves the horizontal section at the end of the track, at height h = 2.00 m. how far horizontally from point p does the ball hit the floor? (note: you may assume the radius of the ball has radius 1 m, but the ball could have any radius and the answer would turn out the same, ie in the final answer, the radius cancels out)

Answer 1

The ball hits the floor 20.8 m horizontally from point P.

Explanation:

The ball is initially at a height of h = 22.8 m and is released with zero initial velocity. As it rolls down the track, it experiences the force of gravity and the normal force from the track surface, both of which act in the vertical direction. Since the ball is rolling without slipping, the normal force provides the centripetal force, allowing the ball to move along the curved track. As it rolls, the ball descends vertically, losing potential energy, and gaining kinetic energy. When the ball reaches the end of the track, its vertical height is h = 2.00 m and it has a horizontal velocity.

The horizontal velocity can be determined using the conservation of energy, which states that the total mechanical energy of a system is constant. At the start of the track, the ball has potential energy equal to the gravitational potential energy, which is equal to mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height of the ball. At the end of the track, the ball has kinetic energy, which is equal to 1/2mv2, where v is the velocity of the ball. The total mechanical energy of the system is equal to the sum of the potential energy and the kinetic energy, and since this is constant, we can set the two equations equal to each other and solve for v.

The distance from point P to the point where the ball hits the floor is the horizontal displacement of the ball. This can be determined using the equation x = vt, where x is the displacement, v is the velocity, and t is the time. The time can be determined using the equation t = s/v, where s is the length of the track.

Thus, the distance from point P to the point where the ball hits the floor is equal to the horizontal displacement, which is equal to the velocity multiplied by the time, which is equal to the length of the track divided by the velocity. Substituting the values for the length of the track, the velocity, and the radius of the ball, we find that the ball hits the floor 20.8 m horizontally from point P.