### Case Study 2: Bouncing Ball

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 $h_{\text{initial}}$

### Worked Example 4: Energy Conversion of A Stone

A boy throws a stone into the air and catches it on the way down. State the energy conversions that take place.

**Note:** This will be a slight variation of the Case Study 2: Bouncing Ball.

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- Just after the stone leaves the boy’s hand, the stone has maximum kinetic energy and minimum gravitational potential energy.
- As it rises, it’s kinetic energy is gradually converted into gravitational potential energy.
- At the point of maximum height, the stone has only gravitational potential energy. At this point, the stone has stopped momentarily so the kinetic energy of the stone is zero.
- As it is falling down, the stone’s gravitational potential energy is gradually converted into kinetic energy.
- Just before the stone reaches the boy’s hand, it will have maximum kinetic energy and minimum gravitational potential energy. However, due to energy losses from friction/air resistance, the new kinetic energy of the stone will still be smaller than it’s initial kinetic energy.

### Worked Example 5: Block on Frictionless Slope

A block of mass 5 kg slides from rest (at the top of the slope) through a distance of 30 m down a frictionless slope. What is the kinetic energy of the block at the bottom of the slope?

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Note that the height of the slope is 5 m. At the top of the slope, the block will possess some gravitational potential energy.

The gravitational potential energy of the block at the top of the slope is given by:

$$\begin{aligned} E_{GPE} &= mgh \\ &= 5 \times 10 \times 5 \\ &= 250 \text{ J} \end{aligned}$$

As the block slides down the slope, the gravitational potential energy will be converted into kinetic energy. Hence, when the block reaches the bottom of the slope, ALL the gravitational potential energy will be converted into kinetic energy.

**The kinetic energy of the block at the bottom of the slope will be 250 J.**

### Worked Example 6: Block on a rough Slope

A block of mass 5 kg slides from rest (at the top of the slope) through a distance of 30 m down a rough slope. Considering that 30 J is dissipated as heat due to friction, what is the kinetic energy of the block at the bottom of the slope?

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Notice that the question is the same as the previous worked example except for the presence of friction.

Similarly, the gravitational potential energy of the block at the top of the slope will be 250 J.

As the block slides down the slope, the gravitational potential energy will be converted into kinetic energy AND heat (friction). Hence, when the block reaches the bottom of the slope, ALL the gravitational potential energy will be converted into kinetic energy AND heat (friction).

This means that:

$$\text{Gravitational Potential Energy} = \text{Kinetic Energy} + \text{Heat}$$

Given that 30 J is dissipated as heat, we have:

$$\begin{aligned} E_{kin} &= 250-30 \\ &= 220 \text{ J} \end{aligned}$$

#### Self-Test Question 1

Consider a car moving at a speed of $v \text{ ms}^{-1}$. If the speed of the car slows down to $\frac{v}{2} \text{ ms}^{-1}$, how much will the kinetic energy of the car decrease by?

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Recall that kinetic energy is given by $\frac{1}{2} mv^{2}$.

We will use $E_{i}$ to denote the initial kinetic energy and $E_{f}$ to denote the final kinetic energy.

We have:

$$\begin{aligned} E_{i} &= \frac{1}{2} m v^{2} \\ E_{f} &= \frac{1}{2} m \left( \frac{v}{2} \right)^{2} \end{aligned}$$

Notice that $E_{f}$ can be simplified to:

$$\begin{aligned} E_{f} &= \frac{1}{4} \frac{1}{2} m v^{2} \\ &= \frac{1}{4} E_{i}$

The kinetic energy of the car will decrease by a factor of 4.

### Self-Test Question 2

A pendulum bob swings from one end to the other. At which point (s) will the gravitational potential energy of the pendulum be (a) maximum, (b) minimum?

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Recall that the gravitational potential energy is given by $E_{GPE} = mgh$.

a) The gravitational potential energy of the pendulum will be the highest (maximum) when it is at the maximum height. The pendulum bob is at the maximum height at the two extreme ends of the oscillation.

b) The gravitational potential energy of the pendulum will be the lowest (minimum) when it is at the minimum height. The pendulum bob is at the minimum height at the centre of the oscillation.

Gaga Godfrey John scribbled

Adjust your way of writing the formulars

Farhan Johnson scribbled

But no one is changing it, why?

Nwaniji Samuel scribbled

I think there is an error here: “When the raised pendulum is released from A, it will swing towards the equilibrium position (Point B). During this movement, the gravitational potential energy is converted into kinetic energy. Hence, at A, gravitational potential energy is at minimum, while kinetic energy is at maximum.” The sentence- “Hence, at A,” should be hence at B,

Saad Chaudhary scribbled

Yes, there is an error…

Fring Maustar scribbled

Yes it is written wrong here