# Case Study 2: Energy Conversion for A Bouncing Ball

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Consider a ball with mass $m$ dropped from a height of $h_{\text{initial}}$ m from the ground.

Stage 1: Initially, the ball will be at height $h$ m above the ground and will have the following properties:

• Gravitational Potential Energy of $mgh_{\text{initial}}$
• Kinetic Energy: 0
• Since the ball will be falling, the gravitational potential energy will be converted into kinetic energy + energy dissipated as heat into surroundings (due to friction/air resistance)

Stage 2: Just before the ball touches the ground, the ball will have the following properties:

• Gravitational Potential Energy is minimum (~ 0)
• Kinetic Energy is maximum ($\text{G.P.E}-\text{Energy dissipated as heat}$)
• When the ball comes into contact with the ground, some energy will be lost as heat and sound (the sound of the ball contacting the ground).

Stage 3: The ball will rebound. Just after the ball rebounds, the ball will have the following properties:

• Gravitational Potential Energy is minimum (~ 0)
• Kinetic Energy is given by ($\text{G.P.E}-\text{Total Energy Dissipated}$)
• Since the ball will be rising, the kinetic energy will be converted into gravitational potential energy + energy dissipated as heat into surroundings (due to friction/air resistance).

Stage 4: The ball reaches maximum height, $h_{\text{max}}$. At this point, the ball will have the following properties:

• Gravitational Potential Energy of $mgh_{\text{max}}$
• Kinetic Energy: 0
• Since energy is lost (due to friction/air resistance and at the bounce), the new maximum height ($h_{\text{max}}$) will be lower than the initial maximum height ($h_{\text{initial}}$).
• This means that the new G.P.E is smaller than the inital G.P.E.
• The ball will never reach the height $h_{\text{initial}}$ 