A skateboarder approaches the circular hill with a velocity of 8.0 m/s at point B. The angle formed between the lines that join point AO and BO is 20 degrees.

a) what is her speed at the top of the hill(Point A) if she coasts up to this point?
b) what min speed must she have at B to coast up to the hill and still reach the top?

To answer these questions, we need to understand the physics principles of conservation of energy and circular motion.

a) To find the skateboarder's speed at the top of the hill (Point A), we can use the conservation of energy. At Point A, all the skateboarder's initial kinetic energy will be converted to potential energy.

The formula for kinetic energy (KE) is given by:
KE = (1/2) * mass * velocity^2

The formula for potential energy (PE) is given by:
PE = mass * g * height

Where mass is the skateboarder's mass, velocity is the skateboarder's velocity, g is the acceleration due to gravity, and height is the height of the hill.

Since the skateboarder is coasting up the hill, she will reach a maximum height when all her kinetic energy is converted to potential energy. Therefore, the initial kinetic energy will be equal to the potential energy at the top of the hill.

Equating the two energies, we have:
(1/2) * mass * velocity^2 = mass * g * height

We can cancel the mass from both sides of the equation:
(1/2) * velocity^2 = g * height

Solving for velocity, we get:
velocity^2 = 2 * g * height

Taking the square root of both sides, we find:
velocity = sqrt(2 * g * height)

Substituting the given values of g (acceleration due to gravity) and height, we can calculate the skateboarder's speed at the top of the hill.

b) To find the minimum speed the skateboarder must have at Point B in order to coast up the hill and still reach the top (Point A), we need to consider the forces acting on the skateboarder.

When the skateboarder is at the top of the hill, the net force acting on her is equal to zero. This means that the centripetal force keeping her moving in a circular path is balanced by the gravitational force pulling her downward.

The formula for the centripetal force is given by:
Fc = mass * velocity^2 / radius

The gravitational force is given by:
Fg = mass * g

Since the centripetal force and the gravitational force are equal, we can set them equal to each other:
mass * velocity^2 / radius = mass * g

We can cancel the mass from both sides of the equation:
velocity^2 / radius = g

Solving for velocity, we get:
velocity = sqrt(radius * g)

Substituting the given values of the angle formed between the lines and the radius of the hill, we can calculate the minimum speed the skateboarder must have at Point B to coast up the hill and still reach the top.