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## End of chapter exercises

Give one word/term for the following descriptions.

1. The force with which the Earth attracts a body.

2. The unit for energy.

3. The movement of a body in the Earth's gravitational field when no other forces act on it.

4. The sum of the potential and kinetic energy of a body.

5. The amount of matter an object is made up of.

a) Gravitational force

b) Joules

c) Free fall

d) Mechanical energy

e) Mass

Consider the situation where an apple falls from a tree. Indicate whether the following statements regarding this situation are TRUE or FALSE. Write only “true” or “false”. If the statement is false, write down the correct statement.

1. The potential energy of the apple is a maximum when the apple lands on the ground.

2. The kinetic energy remains constant throughout the motion.

3. To calculate the potential energy of the apple we need the mass of the apple and the height of the tree.

4. The mechanical energy is a maximum only at the beginning of the motion.

5. The apple falls at an acceleration of 9,8 m·s−2 .

a) False. The kinetic energy of the apple is a maximum when the apple lands on the gorund.

b) False. The mechanical energy remains constant throughout it's motion.

c) True. However, we must also choose a reference point.

d) False. The kinetic energy is a maximum only at the beginning of the motion.

e) True.

A man fires a rock out of a slingshot directly upward. The rock has an initial velocity of 15 m·s−1 .

1. What is the maximum height that the rock will reach?

2. Draw graphs to show how the potential energy, kinetic energy and mechanical energy of the rock changes as it moves to its highest point.

a)

 $K{E}_{b}ottom=\frac{1}{2}m{v}^{2}=P{E}_{t}op$ $\frac{1}{2}m{v}^{2}=m\text{g}h$ $h=\frac{{v}^{2}}{2\text{g}}$ $h=\frac{{15}^{2}}{2\left(9,8\right)}$ $h=11,48m$

b) Potential energy:

Kinetic energy:

Mechanical energy:

A metal ball of mass 200 g is tied to a light string to make a pendulum. The ball is pulled to the side to a height (A), 10 cm above the lowest point of the swing (B). Air friction and the mass of the string can be ignored. The ball is let go to swing freely.

1. Calculate the potential energy of the ball at point A.

2. Calculate the kinetic energy of the ball at point B.

3. What is the maximum velocity that the ball will reach during its motion?

We first convert all units:

mass: $\frac{200}{1000}=0,2kg$

Height: $\frac{10}{100}=0,1m$

We also draw a rough sketch:

a) $PE=mgh=\left(0,2\right)\left(9,8\right)\left(0,1\right)=0,196J$

b) By conservation of mechanical energy:

 $P{E}_{A}=K{E}_{B}$ $K{E}_{B}=0,196J$

c)

 $\frac{1}{2}m{v}^{2}=0,196$ $\frac{1}{2}\left(0,2\right){v}^{2}=0,196$ $\left(0,1\right){v}^{2}=0,196$ $v=1,4m\cdot {s}^{-1}$

A truck of mass 12 tons is parked at the top of a hill, 150 m high. The truck driver lets the truck run freely down the hill to the bottom.

1. What is the maximum velocity that the truck can achieve at the bottom of the hill?

2. Will the truck achieve this velocity? Why/why not?

a) 1 ton = 1 000 kg so mass = 1 200 kg.

$PE=mgh=\left(1200\right)\left(9,8\right)\left(150\right)=1764000J$

From the conservation of mechanical energy:

 $KE=\frac{1}{2}m{v}^{2}=1764000$ $\frac{1}{2}\left(1200\right){v}^{2}=1764000$ $v=54,22m\cdot {s}^{-1}$

b) No. The above calculations do not take friction into account and this will slow the truck down.

A stone is dropped from a window, 6 m above the ground. The mass of the stone is 25 g. Use the Principle of Conservation of Energy to determine the speed with which the stone strikes the ground.

We must first convert the mass to kg:

$m=\frac{25}{1000}=0,025kg$

a)

From the conservation of mechanical energy:

 $P{E}_{t}op=K{E}_{b}ottom$ $mgh=\frac{1}{2}m{v}^{2}$ ${v}^{2}=2gh$ $v=\sqrt{2}\left(9,8\right)\left(3\right)\right)$ $v=7,7m\cdot {s}^{-1}$

So our answer from part a is indeed correct.