Explain why the skier's speed increases as their gravitational potential energy decreases.

Sure, let's break it down step by step:

1. Gravitational Potential Energy (GPE):

   • GPE is the energy an object possesses due to its position in a gravitational field. For a skier at the top of a hill, this energy is given by the formula $\text{GPE} = mgh$, where $m$ is the mass of the skier, $g$ is the acceleration due to gravity, and $h$ is the height above the ground.

2. Conversion of Energy:

   • As the skier moves downhill, they lose height ($h$ decreases). According to the law of conservation of energy, energy cannot be created or destroyed, only transformed from one form to another.

3. Decrease in GPE:

   • As the skier descends, their gravitational potential energy decreases because $h$ is getting smaller.

4. Increase in Kinetic Energy (KE):

   • The decrease in gravitational potential energy is converted into kinetic energy (KE), which is the energy of motion. The formula for kinetic energy is $\text{KE} = \frac{1}{2}mv^2$, where $m$ is the mass of the skier and $v$ is their velocity.

5. Relationship Between GPE and KE:

   • As the skier's GPE decreases, this energy is transferred into KE. Since the skier's mass $m$ remains constant, an increase in KE means an increase in the skier's velocity $v$.

6. Conclusion:

   • Therefore, as the skier's gravitational potential energy decreases, their kinetic energy increases, resulting in an increase in speed. This is why the skier's speed increases as they descend the hill and their gravitational potential energy decreases.