1. A student lifts his 2.0-kg pet rock 2.8m straight up. He then lets it drop to the ground. Use the Law of Conservation of Energy to calculate how fast the rock will be moving .....

a. half way down
b. just before it hits the ground

2. A 65-kg girl is running with a speed of 2.5 m/s. How much kinetic energy does she have? She grabs on to a rope that is hanging from the ceiling, and swings from the end of the rope. How high off the ground will she swing?

3. How much work must be done to increase the speed of a 12-kg bicycle ridden by a 68-kg rider from 8.2 m/s to 12.7 m/s?

To solve these questions, we will need to make use of various principles and equations from physics. Let me break down the steps for you:

1. Question 1:
a. To calculate the speed of the rock at the halfway point, we can use the concept of conservation of energy. At the halfway point, the rock will have lost potential energy equal to the work done against gravity, and gained an equal amount of kinetic energy.
- Calculate the potential energy of the rock at its initial height using the formula: Potential Energy = mass * gravity * height.
- Divide the potential energy by 2 to find the potential energy at the halfway point.
- Set this potential energy equal to the kinetic energy at that point using the formula: Kinetic Energy = (1/2) * mass * velocity^2.
- Solve for the velocity at the halfway point.

b. To calculate the speed just before the rock hits the ground, again we can use the law of conservation of energy.
- Set the potential energy at the top of its initial position equal to the kinetic energy just before it hits the ground.
- Use the same formula for potential energy and the formula for kinetic energy to solve for the velocity just before impact.

2. Question 2:
- Calculate the kinetic energy of the girl using the formula: Kinetic Energy = (1/2) * mass * velocity^2. Substituting the given values, solve for the kinetic energy.

- To find the height the girl will swing to, we can apply the principle of conservation of mechanical energy. At the highest point of her swing, her kinetic energy will be zero and all her initial kinetic energy will have been converted to potential energy.
- Set the initial kinetic energy equal to the potential energy at the highest point using the same formulas for kinetic and potential energy.
- Solve for the height above the ground.

3. Question 3:
- Calculate the initial kinetic energy of the bicycle and rider using the formula: Kinetic Energy = (1/2) * mass * velocity^2. Substituting the given values, solve for the initial kinetic energy.
- Calculate the final kinetic energy using the same formula.
- The work done to increase the speed is equal to the change in kinetic energy.
- Subtract the initial kinetic energy from the final kinetic energy to find the work required.

These step-by-step instructions should help you solve the given physics problems.