Work, Power and Energy

Lessons

Senior 1 · Physics

Work, Power and Energy

5 lessons

Lesson 4 of 5

Kinetic and Potential Energy

25 min

Speed and height

On the Huye–Gicumbi stage of the Tour du Rwanda, riders climb for kilometres at walking pace, then race down the other side faster than the cars. At the top of the hill they have energy because of their height; on the way down it becomes energy of motion. The same two kinds of energy explain why a mango from the top of the tree hurts more than one from a low branch, and why a crash at 90 km per hour is so much worse than one at 60.

Definition

Kinetic energy ( $E_{k}$ ) is the energy a body has because it is moving. For a body of mass $m$ moving with speed $v$ :

E_k = \frac{1}{2}mv^2

with $m$ in kg, $v$ in $\mathrm{m\,s^{-1}}$ and $E_{k}$ in joules. Example: a 2.0 kg ball at $3.0\,\mathrm{m\,s^{-1}}$ has $E_k = \frac{1}{2} \times 2.0 \times 3.0^2 = 9.0\,\mathrm{J}$ .

Definition

Gravitational potential energy ( $E_{p}$ ) is the energy a body has because of its height above a chosen reference level (usually the ground). It equals the work done lifting it there:

E_{p} = m g h

with $m$ in kg, $g = 9.8\,\mathrm{N\,kg^{-1}}$ and the height $h$ in metres. Example: a 2.0 kg bag on a 1.5 m shelf has $E_p = 2.0 \times 9.8 \times 1.5 = 29.4\,\mathrm{J}$ relative to the floor.

Experiment: craters in the sand

You need: a tray of fine damp sand, two steel balls of different mass, a metre rule clamped upright, and a ruler to measure crater depth.

Hold the lighter ball beside the rule at a height of 0.20 m above the sand and let it go (do not throw it).
Measure the depth of the crater. Smooth the sand flat.
Repeat from 0.40 m, 0.60 m and 0.80 m, then repeat with the heavier ball.

Figure 4.1: The higher the ball starts, the more potential energy it has, the faster it lands and the deeper the crater it makes.

Height $h$ (m)	Crater depth, 50 g ball (mm)	Crater depth, 100 g ball (mm)
0.20	6	9
0.40	9	13
0.60	11	16
0.80	13	18

Conclusion: the crater gets deeper as the height or the mass increases. Both increase $m g h$ , the potential energy at the start, which becomes kinetic energy by the time the ball hits the sand and then does work pushing the sand aside.

Important

Safety. Keep hands and faces clear of the drop line. Use damp sand so it does not fly up into eyes, and never drop a heavy ball from more than a metre.

Worked example 1: a racing cyclist

A rider and bicycle have a total mass of 75 kg. Find their kinetic energy at $12\,\mathrm{m\,s^{-1}}$ , and at twice that speed.

E_k = \frac{1}{2}mv^2 = \frac{1}{2} \times 75 \times 12^2 = 5400\,\mathrm{J}

E_k = \frac{1}{2} \times 75 \times 24^2 = 21\,600\,\mathrm{J}

Doubling the speed multiplies the kinetic energy by $2^{2} = 4$ .

Figure 4.2: Kinetic energy of a 75 kg cyclist. Because the speed is squared, the curve gets steeper and steeper: from 6 to 12 metres per second the energy goes up four times.

Worked example 2: a falling mango

A 150 g mango hangs 6.0 m above the ground. Find its gravitational potential energy relative to the ground.

Convert first: $m = 150\,\mathrm{g} = 0.15\,\mathrm{kg}$ .

E_p = mgh = 0.15\,\mathrm{kg} \times 9.8\,\mathrm{N\,kg^{-1}} \times 6.0\,\mathrm{m} = 8.82\,\mathrm{J} \approx 8.8\,\mathrm{J}

Worked example 3: finding a speed from kinetic energy

A 0.50 kg football has 100 J of kinetic energy. How fast is it moving?

E_k = \frac{1}{2}mv^2 \quad\Rightarrow\quad v^2 = \frac{2E_k}{m} = \frac{2 \times 100}{0.50} = 400

v = \sqrt{400} = 20\,\mathrm{m\,s^{-1}}

Warning

Common mistakes

Square only the speed: $\frac{1}{2}mv^2$ is $\frac{1}{2} \times m \times (v \times v)$ , not $\left(\frac{1}{2}mv\right)^2$ .
Convert speeds to $\mathrm{m\,s^{-1}}$ : $54\,\mathrm{km\,h^{-1}} = \frac{54\,000\,\mathrm{m}}{3600\,\mathrm{s}} = 15\,\mathrm{m\,s^{-1}}$ .
Convert grams to kilograms, and use the vertical height for $E_{p}$ .

Speed on Rwanda's roads

Because kinetic energy depends on $v^{2}$ , a small rise in speed makes a big rise in the energy the brakes must remove, or that a crash must absorb.

Speed of a 10 000 kg bus	In $\mathrm{m\,s^{-1}}$	Kinetic energy
30 km per hour	8.3	0.35 MJ
60 km per hour	16.7	1.39 MJ
90 km per hour	25.0	3.13 MJ

Going from 60 to 90 km per hour gives the bus 2.25 times as much kinetic energy. This is one reason why Rwanda requires buses and lorries to carry speed governors, and why speed limits near schools are low.

Tip

Quick check: twice the mass, twice the kinetic energy; twice the speed, FOUR times the kinetic energy; twice the height, twice the potential energy.

Energy intro: kinetic and potential (Khan Academy India). Watch for the two formulas and for how the height is measured from a chosen reference level.

✦

Practice

Write the formulas for kinetic energy and gravitational potential energy, with the unit of each quantity.
Find the kinetic energy of a 4.0 kg ball moving at $5.0\,\mathrm{m\,s^{-1}}$ .
Find the potential energy gained by a 60 kg pupil who climbs 10 m up a hill.
A 1200 kg car travels at 72 km per hour. Convert the speed to $\mathrm{m\,s^{-1}}$ and find its kinetic energy.
A body's speed is tripled. By what factor does its kinetic energy change?
A 250 g bird flies at a height of 20 m. Find its potential energy relative to the ground.
A 2.0 kg stone has 36 J of kinetic energy. Find its speed.
To what height must a 5.0 kg box be lifted to gain 245 J of potential energy?

Show answersHide answers

$E_k = \frac{1}{2}mv^2$ and $E_{p} = m g h$ : $m$ in kg, $v$ in $\mathrm{m\,s^{-1}}$ , $g$ in $\mathrm{N\,kg^{-1}}$ , $h$ in m, energies in J.
$E_k = \frac{1}{2} \times 4.0 \times 25 = 50\,\mathrm{J}$
$E_p = 60 \times 9.8 \times 10 = 5880\,\mathrm{J}$
$72\,\mathrm{km\,h^{-1}} = 20\,\mathrm{m\,s^{-1}}$ ; $E_k = \frac{1}{2} \times 1200 \times 400 = 240\,000\,\mathrm{J}$
$3^{2} = 9$ : it becomes nine times as large.
$E_p = 0.25 \times 9.8 \times 20 = 49\,\mathrm{J}$
$v^2 = \frac{2 \times 36}{2.0} = 36$ , so $v = 6.0\,\mathrm{m\,s^{-1}}$
$h = \frac{E_p}{mg} = \frac{245}{5.0 \times 9.8} = 5.0\,\mathrm{m}$

Key points

Kinetic energy: $E_k = \frac{1}{2}mv^2$ . Doubling the speed quadruples it.
Gravitational potential energy: $E_{p} = m g h$ , measured from a chosen level, using the vertical height.
Together, kinetic and potential energy make up mechanical energy.
Always convert to kg, $\mathrm{m\,s^{-1}}$ and m before substituting.

Sources and credits

7.2: Kinetic Energy and the Work-Energy Theorem, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Consulted for the physics; no text was copied.
7.3: Gravitational Potential Energy, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Consulted for the physics; no text was copied.
Photo: Huye - Gicumbi, Stage 3 of TOUR DU RWANDA.jpg by Tuyishimire Emmanuel, Wikimedia Commons, licensed under CC BY-SA 4.0.
Video: Energy intro (kinetic & potential) | Work & Energy | Physics | Khan Academy, Khan Academy India - English, embedded from YouTube.

Written by Ganzaa: the lesson text, the sand-crater experiment and its sample results, the diagrams, the road-speed table, worked examples and practice exercises.

1Speed and height

2Experiment: craters in the sand

3Speed on Rwanda's roads

Speed and height

Experiment: craters in the sand

Speed on Rwanda's roads