Final answer:
To determine the vertical height of the hill, we can use the principle of conservation of mechanical energy. The initial kinetic energy at the bottom of the hill is equal to the final potential energy at the top of the hill. By setting the initial kinetic energy equal to the final potential energy, we can solve for the height of the hill.
Step-by-step explanation:
To determine the vertical height of the hill, we can use the principle of conservation of mechanical energy. At the bottom of the hill, the bicyclist has only kinetic energy. At the top of the hill, the bicyclist has only potential energy. Therefore, the initial kinetic energy at the bottom of the hill is equal to the final potential energy at the top of the hill.
The formula for kinetic energy is given by K = 1/2 mv^2, where m is the mass and v is the velocity. The formula for potential energy is given by P = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
The kinetic energy at the bottom of the hill is given by K = 1/2 mv^2 = 1/2 * 1 kg * 20.0 m/s * 20.0 m/s = 200 J.
The potential energy at the top of the hill is given by P = mgh, where h is the height we need to find. Since the bicyclist comes to a stop at the top of the hill, the final velocity v is 0 m/s. Therefore, the equation becomes P = mgh = 1 kg * 9.8 m/s^2 * h = 9.8h J.
Since the initial kinetic energy is equal to the final potential energy, we can set 200 J = 9.8h J and solve for h:
200 J = 9.8h J
h = 200 J / 9.8 J = 20.4 m.
Therefore, the vertical height of the hill is 20.4 m.