91Ó°ÊÓ

Potential energy is the stored energy of an object due to its position or height. In the context of our problem, the golf ball starts at the top of the stairs with a certain amount of potential energy. This energy is determined by the formula \( PE = mgh \), where:

• \( m \) is the mass of the golf ball,
• \( g \) is the acceleration due to gravity, typically \( 9.81 \, \text{m/s}^2 \),
• \( h \) is the height above the ground, which is \( 3 \, \text{meters} \) in this case.

At the top of the stairs, the ball has maximum potential energy because it is elevated. As it begins to bounce down the steps, this potential energy will be converted into other forms of energy.

Kinetic energy is the energy of motion. When the golf ball starts descending the stairs, its potential energy at the top converts into kinetic energy. The formula for kinetic energy is \( KE = \frac{1}{2} mv^2 \), where:

• \( m \) is the mass of the ball,
• \( v \) is the velocity of the ball.

As the ball falls, it speeds up, increasing its kinetic energy due to gravity. At the bottom of the stairs, all the initial potential energy has been transformed into kinetic energy. This means the ball is moving at its fastest, having no potential energy left as it reaches ground level. The conversion process is efficient because the collisions along the stairs are elastic, meaning no energy is lost to other forms like heat or sound. Thus, right before reaching the bottom, the ball moves with a velocity calculated from \( v = \sqrt{2gh} \).

Energy conservation is a key principle, especially in elastic collisions. It states that the total energy in a closed system remains constant over time. In our golf ball problem, the potential and kinetic energies are interchangeable forms of mechanical energy.
Since the collisions with the stairs are elastic, the energy isn't lost to external factors. Instead, it just changes form from potential to kinetic and possibly back, depending on the bounce.

• At the top of the stairs: Maximum potential energy, zero kinetic energy.
• At the bottom of the stairs: Zero potential energy, maximum kinetic energy.
• Upon bouncing up: Kinetic energy converts back into potential energy.

This conservation law ensures that the ball can bounce back to its original height after reaching the bottom of the stairs. The kinetic energy at the bottom allows the ball to reach its initial potential energy level as it ascends, resulting in a bounce height of \( 3.00 \, \text{m} \). This is because the total mechanical energy remains unchanged throughout the motion, aligning perfectly with the conservation law.