Where Does the Energy Go When a Ball Bounces?: Going Further 9–12

Energy you can count

In Grade 9 you met work: when a force moves something, work = force × distance moved, measured in joules (J). One joule is about the work you do lifting a 100 g apple (a weight of about 1 N) up by 1 m. Work is how energy moves from one thing to another: when you lift the apple, energy goes from you into the apple and the Earth. In a machine, the input work is the energy you put in, and the output work is the energy the machine passes on Grade 9: Efficiency.

So energy is something you can count, like money in a bank account. It moves from one account to another, and changes from one kind to another, but if you add up every account, the total never changes. To check that, you need a way to count each kind. Physicists sort all the kinds into two big families:

Energy of motion, called kinetic energy: a moving bus, a flying cricket ball, the jiggling molecules of hot tea.
Energy of position, called potential energy: a ball held up high, a stretched catapult band, the electrons and atoms in a molecule of petrol. It is energy stored because things that push or pull on each other are held at a distance from each other.

In Grade 6 you met these two as kinetic energy and potential energy. Now you will put numbers on them.

A 1 kg bag of rice is lifted from the floor onto a shelf 1.5 m high. About how much work is done? (A 1 kg mass weighs about 9.8 N.)

Pick an answer to see why.

Kinetic energy: the energy of motion

How much energy does a moving thing have? Find out how much work it takes to get it moving.

Push a trolley of mass m, starting from rest, with a steady force F along a smooth, level floor. You learned in Grade 9 that the force gives it an acceleration a = F ÷ m Grade 9: Newton’s second law, and that after it has moved a distance s its velocity v is given by v² = u² + 2as, with u = 0 Grade 9: Equations of linear motion. Put them together:

$\text{work done} = F \times s = (ma) \times \frac{v^2}{2a} = \tfrac{1}{2}mv^2$

The acceleration cancels out. However hard or gently you push, the work it takes to bring a mass m up to a speed v is always the same. That work is now stored in the trolley’s motion, and it comes out again when the trolley is stopped: it can squash a box, or heat the brakes. So the kinetic energy (KE) of a body of mass m moving at speed v is

$\text{KE} = \tfrac{1}{2}mv^2$

with m in kg and v in m/s, giving joules.

Look at what the formula says. Kinetic energy is in proportion to the mass: a loaded truck has twice the kinetic energy of the same truck empty, at the same speed, if the load weighs as much as the truck. But it goes with the square of the speed: at twice the speed, a bus has four times the kinetic energy, and its brakes have four times as much energy to get rid of. That is why a crash at 80 km/h is so much worse than one at 40 km/h.

Worked example

A fast bowler’s delivery

A cricket ball has a mass of 160 g. A fast bowler bowls it at 144 km/h (40 m/s). How much kinetic energy does it have? Compare it with a bus of 10,000 kg moving at 54 km/h (15 m/s).

Answer

Cricket ball: m = 0.16 kg, v = 40 m/s.
$\text{KE} = \tfrac{1}{2}mv^2 = \tfrac{1}{2} \times 0.16\ \text{kg} \times (40\ \text{m/s})^2 = 0.08 \times 1600 = 128\ \text{J}$
Bus: m = 10,000 kg, v = 15 m/s.
$\text{KE} = \tfrac{1}{2} \times 10{,}000 \times 15^2 = 5000 \times 225 = 1{,}125{,}000\ \text{J} \approx 1.1\ \text{MJ}$
The bus, slower but over 60,000 times heavier, has nearly 9,000 times as much kinetic energy. (Units: kg × (m/s)² = kg m²/s², which is the same as N m, the joule.)

A car’s speed goes up from 30 km/h to 90 km/h. Its kinetic energy becomes

Pick an answer to see why.

Gravitational potential energy: the energy of height

Now lift something. To raise a mass m steadily, you must push up with a force equal to its weight, mg Grade 10: Weight. Lifted through a height h, the work you do is force × distance:

$\text{work done} = mg \times h$

That work isn’t lost. Let go, and the falling mass gives it back as speed. While it is held up, the energy is stored, as gravitational potential energy (GPE):

$\text{GPE} = mgh$

with m in kg, g = 9.8 m/s² and h in meters, giving joules.

Two things to notice. First, h is the height above a level you choose: the floor, the table, the ground outside. What matters in every calculation is how much the GPE changes, and the change is the same whichever level you call zero. Second, the energy isn’t really “in” the ball. It belongs to the ball and the Earth together: it is stored because they pull on each other and have been pulled apart. Physicists say it is stored in the gravitational field between them (you met fields in Why Is Gravity So Weak?). A ball far out in space, with nothing to fall towards, would have no gravitational potential energy to give.

A ball on a table is 1 m above the floor and 4 m above the ground outside the window. Its GPE is

Pick an answer to see why.

Swapping one for the other

Drop a ball. As it falls, it loses height and gains speed: gravitational potential energy is turning into kinetic energy. Is the swap exact?

You can check with the equation you used for falling bodies in Grade 10 Grade 10: Equations of motion for free fall. A body dropped from rest (u = 0) that has fallen a height h has v² = 2gh. Multiply both sides by ½m:

$\tfrac{1}{2}mv^2 = mgh$

Kinetic energy gained = gravitational potential energy lost, exactly. While nothing but gravity acts, GPE + KE stays the same all the way down. Here is the swap, worked out for the tennis ball dropped from 2.54 m that you will meet again below (mass 58 g, so its GPE at the top is 0.058 × 9.8 × 2.54 = 1.44 J):

Height (m)	Speed (m/s)	GPE (J)	KE (J)	GPE + KE (J)
2.54 (let go)	0	1.44	0	1.44
1.91	3.53	1.08	0.36	1.44
1.27	4.99	0.72	0.72	1.44
0.64	6.11	0.36	1.08	1.44
0 (hits the floor)	7.06	0	1.44	1.44

This is the principle of conservation of energy put to work: if you know how the energy changes in one part of a system, you can work out the change in another part, without knowing anything about the forces or the path. Energy gives you a short cut.

Worked example

The Dashain swing

At Dashain, villages put up a tall bamboo swing, the linge ping. At the top of a big swing, a child is 3.0 m higher than at the lowest point. How fast is the child going at the bottom? (Ignore air resistance and friction.)

Answer

GPE lost on the way down = KE gained at the bottom:
$mgh = \tfrac{1}{2}mv^2$
The mass m is on both sides, so it cancels: a heavy child and a light one reach the same speed.
$v = \sqrt{2gh} = \sqrt{2 \times 9.8\ \text{m/s}^2 \times 3.0\ \text{m}} = \sqrt{58.8} = 7.7\ \text{m/s}$
That is about 28 km/h, as fast as a bicycle going quickly. The rope makes the child go round a curve rather than straight down, but the rope pulls at right angles to the motion, so it does no work: only the drop in height counts.

Four tall bamboo poles, their leafy tops left on, leaning together in two crossed pairs; a child standing on the swing's seat flies high out to one side, against a sunset over hills — **Figure 10.1** A *linge ping* at Dashain, in Palpa. On every swing, energy changes from gravitational to kinetic and back. Photo: Mithun Kunwar, CC BY 2.0

Two stones, one of 1 kg and one of 5 kg, are dropped together from a bridge 20 m high. Ignoring air, which is true when they reach the water?

Pick an answer to see why.

Where the energy goes in a bounce

Now watch the bounce itself. A tennis ball hits the floor at 7 m/s. In a few thousandths of a second it is squashed flat on one side, stops, and springs back up.

At the moment it is squashed most, the ball isn’t moving at all. Its kinetic energy is now stored in the squashed rubber and the squeezed air inside, the way energy is stored in a stretched catapult band. This is elastic potential energy, and you met the idea behind it in Grade 9: an elastic body that is deformed has a restoring force that pushes it back Grade 9: Elasticity and plasticity. As the ball springs back, the elastic energy turns back into kinetic energy, and the ball leaves the floor.

But not all of it. As the rubber squashes and springs back, its long, tangled molecules slide and rub past one another, and the floor shakes a little. That rubbing turns part of the energy into thermal energy: the ball and the floor get very slightly warmer. A very little goes off as the sound of the bounce, and that too ends up as thermal energy in the air. So the ball leaves the floor slower than it arrived, and rises less high.

Many images of the same orange toy basketball on a black background, tracing two arches from left to right, the second lower than the first — **Figure 10.2** A bouncing ball (a toy basketball), photographed with a light flashing 25 times a second, so the images are a twenty-fifth of a second apart. Each arch is lower than the one before: at each bounce, some energy has become thermal energy. Photo: MichaelMaggs Edit by Richard Bartz, CC BY-SA 3.0

How much is lost? Sports’ governing bodies test their balls by exactly this experiment. Each ball must be dropped from a fixed height onto a hard surface and bounce back to a height inside a fixed range:

Ball	Mass	Dropped from	Must bounce back to	Middle of range ÷ drop
Tennis ball (ITF)	56.0–59.4 g	254 cm, onto concrete	135–147 cm	141 ÷ 254 = 56%
Football (FIFA Quality)	410–450 g	200 cm, onto a steel plate	125–155 cm	140 ÷ 200 = 70%
Table tennis ball (ITTF)	2.7 g	30.5 cm, onto a steel block	24–26 cm	25 ÷ 30.5 = 82%

Because GPE = mgh, the ratio of the bounce height to the drop height is the fraction of the energy that the ball keeps in each bounce: the bounce’s efficiency, in Grade 9’s words (output ÷ input). A tennis ball keeps about 56% of its energy and turns about 44% into thermal energy. (A little is also lost to air resistance on the way down and up, but over a fall of 2.5 m that is small.)

Figure 10.3 Where the energy goes when a tennis ball is dropped from 254 cm and bounces back to 141 cm (the middle of the ITF’s range). Left: the test, drawn to scale, the ball included. Right: the energy, every band’s width in proportion to the joules it carries.

Worked example

Following the tennis ball’s energy

A tennis ball of mass 58 g is dropped from 2.54 m onto concrete and bounces back to 1.41 m. a. How much energy did it have at the start? b. How much at the top of the bounce? c. How much became thermal energy? d. If all of that stayed in the ball, how much warmer would the ball get? (Take the ball’s specific heat capacity as about 2,000 J/(kg °C), roughly that of rubber.)

Answer

a.GPE at the start = mgh = 0.058 × 9.8 × 2.54 = 1.44 J.
b.GPE at the top of the bounce = 0.058 × 9.8 × 1.41 = 0.80 J.
c.By conservation of energy, the energy that is not there any more as GPE (or KE) has become thermal energy: 1.44 − 0.80 = 0.64 J.
d.Using the heat equation from Grade 10 Grade 10: Calculating heat, Q = msΔθ:

$\Delta\theta = \frac{Q}{ms} = \frac{0.64\ \text{J}}{0.058\ \text{kg} \times 2000\ \text{J/(kg °C)}} = 0.0055\ °\text{C}$
About half of one hundredth of a degree, far too little to feel, and some of the energy goes into the floor, not the ball. One bounce can’t warm a ball. A squash ball hit hard dozens of times can.

Pick a ball and drop it. The ball is dropped from the height in its official test, and keeps the share of its energy that the test says. Watch the bars: the gravitational, kinetic, elastic and thermal energy change all the time, but the total doesn’t. (The bounce, which really lasts a few thousandths of a second, is shown slowed right down.)

Ball:

Height of the ball’s bottom: 2.54 m
Speed: 0.00 m/s
Energy kept at each bounce: 56%

The ball is held still: all its energy is gravitational. Press Drop.

A ball is dropped from 2.0 m and bounces to 1.5 m. What fraction of its energy became thermal energy in the bounce?

Pick an answer to see why.

Every kind of energy is motion or position

Where is the thermal energy, really? Zoom in on the ball.

The ball is made of countless molecules. Before the bounce, they were all moving downwards together at 7 m/s: that shared, orderly motion is the ball’s kinetic energy. After the bounce, the ball as a whole is moving more slowly. The missing energy hasn’t vanished: it is now spread among the molecules as random motion, each jiggling a tiny bit faster than before, in every direction. In Grade 10 you learned that the total kinetic energy of the molecules is a body’s thermal energy, and that temperature measures their average kinetic energy Grade 10: Thermal energy, heat and temperature. So the ball’s energy is still energy of motion. It has only gone from the motion of the whole ball to the motion of its separate molecules.

There is a little more to thermal energy than that. In a solid or a liquid, molecules pull and push on their neighbors, like balls joined by springs. As they jiggle, they keep stretching and squeezing those springs, so part of a body’s thermal energy is the energy of position of its molecules. That is where the energy goes when ice melts: the molecules are pulled out of their fixed places, against the pull of their neighbors, and their energy of position goes up while the temperature stays at 0 °C (more in Why Does Salt Melt at 801 °C but Ice at 0 °C?).

Follow any kind of energy far enough down and you find the same two families again:

Elastic energy in a squashed ball or a stretched catapult band is the energy of position of its atoms, pushed together or pulled apart against the electric forces that hold them in place.
Chemical energy in petrol, food or a battery is energy of position too: of electrons and atoms, held in the arrangement of the molecules. A reaction that rearranges them can release it (Why Do Some Reactions Get Hot?).
Electrical energy stored between two charged metal plates, one positive and one negative, is energy of position as well. The plates pull on each other; to pull them farther apart you must do work, and that work is stored in the electric field between them, just as the work of lifting a ball is stored in the gravitational field between the ball and the Earth.
Sound is the motion of air molecules, squeezed together and spread apart as the wave passes: motion and position, back and forth.

Figure 10.4 At every scale, energy is energy of motion (kinetic) or energy of position, stored in a field between things that push or pull on each other (potential).

So the answer to the question in the title is: the energy of the bouncing ball goes into the motion of molecules, in the ball, the floor and the air. It is all still there; it has just been spread so thinly among so many molecules that you can’t see it, and the temperature rise is too small to feel.

(Light is the one kind that doesn’t fit neatly into either family: it is energy carried by electric and magnetic fields themselves, traveling through space.)

A moving bus brakes to a stop. At the level of molecules, what has happened to its kinetic energy?

Pick an answer to see why.

Proving that heat is energy

A story from science

The brewer who measured heat

In the 1840s, most scientists thought heat was an invisible fluid, called “caloric”, that flowed out of hot things into cold ones. James Prescott Joule, from a family of brewers near Manchester in England, thought instead that heat was a kind of motion, and that a fixed amount of work would always make the same amount of heat.

To test it, he built a copper can of water with a paddle wheel inside, and fixed vanes to stop the water simply turning round with it. Two weights hung on strings wound round the wheel’s axle. As the weights fell, they turned the paddles, which churned the water, and the churning warmed it, very slightly. Joule let the weights fall, wound them up again, and let them fall, again and again, and then measured the water’s rise in temperature: a fraction of a degree, read on thermometers made specially for him, so finely marked that he said he could read them to a two-hundredth of a degree Fahrenheit (less than a hundredth of a degree Celsius).

From the work done by the falling weights (mgh, each time) and the warming of the water, he found how much work makes a given amount of heat. His result, published in 1850, is within 1% of today’s value: it takes about 4.2 J of work to warm 1 g of water by 1 °C. That is why water’s specific heat capacity is 4,200 J/(kg °C), and why the unit of energy is named the joule, after him.

Joule even predicted, in 1845, that the water of the Niagara Falls should be about a fifth of a degree Fahrenheit (about 0.1 °C) warmer at the bottom than at the top, because of its 160-foot fall. William Thomson (later Lord Kelvin) remembered, many years later, meeting Joule on his honeymoon in the Alps in 1847, carrying a long thermometer to measure the water of a waterfall; they found the falling water too broken up into spray to try. Joule’s biographer doubts the honeymoon story, which was first told decades afterwards, but the physics in it is right.

A copper can, open at the top, with a brass paddle wheel and its fixed vanes standing half out of it, on a museum display stand — **Figure 10.5** A replica of the heart of Joule’s apparatus, at the Science and Industry Museum in Manchester: the copper can and the brass paddle wheel, with the fixed vanes that stop the water simply turning round with it. In the experiment the can is full of water, the wheel inside it, and the falling weights turn the paddles. Photo: Chris Allen, CC BY-SA 2.0

Worked example

Warming water by dropping it

Water falls 100 m down a waterfall. If all its gravitational energy ends up as thermal energy in the water, how much warmer is it at the bottom? (Specific heat capacity of water = 4,200 J/(kg °C).)

Answer

Take any mass m of water. The GPE it loses is mgh; the heat that would warm it by Δθ is msΔθ. Set them equal:
$mgh = ms\,\Delta\theta \quad\Rightarrow\quad \Delta\theta = \frac{gh}{s} = \frac{9.8 \times 100}{4200} = 0.23\ °\text{C}$
The mass cancels: a trickle and a river warm by the same amount. Each meter of fall warms water by only 0.0023 °C, which is why Joule needed such good thermometers. For Niagara’s 49 m (160 feet), 0.11 °C: Joule’s fifth of a degree Fahrenheit.

Building an energy model

To follow energy through any change, scientists build an energy model: a set of accounts that must balance. You can do it in four steps.

Choose the system: the things you will keep the accounts for (the ball, the Earth and the floor; a bicycle, its rider and the Earth).
List its energy stores: kinetic, gravitational, elastic, thermal, chemical.
List the energy flowing in or out: work done by something outside (a push, your legs on the pedals), heat flowing in or out, sound or light leaving.
Balance the accounts: the total change in the stores = energy in − energy out. If you know all the changes but one, the last one is the difference.

$\Delta\text{KE} + \Delta\text{GPE} + \Delta(\text{thermal}) + \ldots = \text{energy in} - \text{energy out}$

Each term means something real. A Δ (“change in”) is the final amount minus the starting amount, so it is negative when a store goes down: when the ball falls, ΔGPE is negative and ΔKE is positive, and they add up to zero.

Worked example

Down from Nagarkot

A cyclist free-wheels down the road from Nagarkot (about 2,175 m above sea level) to Bhaktapur (about 1,401 m), braking all the way, and arrives at 10 m/s (36 km/h). The rider and bicycle together have a mass of 75 kg (a made-up but typical figure). a. How much gravitational potential energy is released? b. How much of it is kinetic energy at the bottom? c. Where did the rest go? d. How fast would the bicycle be going if there were no brakes and no air?

Answer

The system is the rider, the bicycle and the Earth. No one pedals, so no energy flows in.

a.The drop is 2,175 − 1,401 = 774 m. GPE released = mgh = 75 × 9.8 × 774 = 569,000 J (569 kJ).
b.KE at the bottom = ½mv² = ½ × 75 × 10² = 3,750 J (3.75 kJ), less than 1% of the energy released.
c.ΔKE + ΔGPE + Δ(thermal) = 0, so Δ(thermal) = −ΔKE − ΔGPE = −3,750 − (−569,000) = 565,000 J. Over 99% of the energy became thermal energy: in the brake blocks and rims, the tyres and the road, and the air pushed out of the way.
d.With all 569 kJ as kinetic energy: v = √(2gh) = √(2 × 9.8 × 774) = 123 m/s, about 440 km/h. Brakes and air are what make the ride possible.

Move the rider down the road, and set the speed. The bars show the energy accounts: the gravitational energy released so far, split into kinetic energy and thermal energy. Try riding the whole way at a steady speed. Then try going fast near the top.

Height dropped: 774 m Speed: 30 km/h

Gravitational energy released, mgh: 568.9 kJ
Kinetic energy, ½mv²: 2.6 kJ
Thermal energy (released − kinetic): 566.3 kJimpossible
Speed with no brakes and no air: 443 km/h

Almost all the energy released has become thermal energy, in the brakes, tyres and air. Only a sliver is kinetic energy.

Activity

Build a model of a falling ball in a spreadsheet

A computational model works out what happens step by step, with a computer doing the sums. You can build one in any spreadsheet (Excel, LibreOffice or Google Sheets).

Columns: time t (s), height h (m), velocity v (m/s), GPE (J), KE (J), total (J).

In the first row, put t = 0, h = 2.54, v = 0, and the ball’s mass, 0.058 kg, in a cell of its own.
In each next row, add 0.02 s to the time. The new velocity is the old velocity − 9.8 × 0.02 (falling, so downwards is negative). The new height is the old height + (new velocity × 0.02).
Work out GPE = mgh and KE = ½mv² in each row, and add them.
Copy the rows down until the height reaches zero. Draw a graph of GPE, KE and the total against time.

Then try: a. Make the time step 0.01 s, then 0.001 s. What happens to the total? b. Add air resistance: take away from the speed a little extra each step, in proportion to v². What happens to the total now, and where is that energy going?

What you should see

The first rows (time step 0.02 s):

t (s)	h (m)	v (m/s)	GPE (J)	KE (J)	Total (J)
0	2.540	0	1.444	0	1.444
0.02	2.536	−0.196	1.442	0.001	1.443
0.04	2.528	−0.392	1.437	0.004	1.442
0.06	2.516	−0.588	1.430	0.010	1.440

The ball reaches the floor after about 0.72 s, at about 7 m/s. The GPE falls along a curve and the KE rises along the mirror of it. The total isn’t perfectly flat: by the last row above the floor it has drifted down to about 1.40 J, because the model uses the speed at the end of each step for the whole step, so the ball falls a little too far in each step. Smaller steps make the drift smaller, which is how you know it comes from the model, not from the physics. With air resistance, the total goes down steadily: energy is leaving the ball as thermal energy in the air.

Activity

Investigate: how bouncy is your ball?

You need: some different balls (a tennis ball, a rubber ball, a football, a table tennis ball, a cricket ball if you have one), a measuring tape, some sticky tape, a wall, and a phone that can record slow-motion video.

Tape the measuring tape to the wall, with zero on the floor.
Hold a ball with its bottom at 100 cm and let it go. Film it in slow motion, and find from the video the height of the bottom of the ball at the top of its first bounce. Do it three times and take the average.
Work out the percentage of its energy each ball keeps: bounce height ÷ drop height × 100%. Which is bounciest? Does the floor matter (tiles, cement, grass, a mat)?
Put a rubber ball in the fridge for an hour, and another in warm sun. Test both. Which bounces higher?

What you should see

A tennis ball keeps a little over half of its energy on a hard floor; a table tennis ball a little more; a cricket ball much less. On grass or a mat, every ball keeps less: the soft floor squashes and doesn’t spring back well, so more energy becomes thermal energy in the floor. The warm ball bounces higher than the cold one, as a squash ball does.

In short

Energy can be counted, in joules. It is never created or destroyed: in any change, the total stays the same, if you count every kind.
Kinetic energy, KE = ½mv²: in proportion to the mass, and to the square of the speed.
Gravitational potential energy, GPE = mgh, stored in the field between a body and the Earth. Only changes in it matter, so you can measure h from any level you choose.
For a falling body, GPE lost = KE gained, so v = √(2gh), whatever its mass.
In a bounce, kinetic energy is stored for an instant as elastic energy, and part of it becomes thermal energy in the ball and floor. Bounce height ÷ drop height is the share of the energy the ball keeps: about 56% for a tennis ball.
At every scale, energy is energy of motion or energy of position (stored in a field). Thermal energy is the random motion of molecules, plus their energy of position.
An energy model balances the accounts: the changes in all the stores add up to the energy flowing in minus the energy flowing out. If you know all but one, you can work out the last.

Questions

A stone falls from a cliff. Halfway down (ignoring air), its kinetic energy is

Pick an answer to see why.

A ball dropped on a hard floor bounces back to 60% of its drop height each time. After two bounces, what share of its starting energy does it still have as GPE at the top?

Pick an answer to see why.

a.

A 1,200 kg car is moving at 20 m/s. a. Find its kinetic energy. b. It brakes to a stop. If all its kinetic energy goes into its four brake discs, each of mass 5 kg and made of iron (s = 460 J/(kg °C)), by how much does each disc warm up?

Answer

a.KE = ½mv² = ½ × 1,200 × 20² = 240,000 J.
b.Each disc gets a quarter: 60,000 J. Δθ = Q ÷ (ms) = 60,000 ÷ (5 × 460) = 26 °C. (In a real car the front brakes do more of the work and some heat goes into the air and the pads, but a hard stop really does warm the discs by tens of degrees. Brakes are machines for turning kinetic energy into thermal energy.)

b.

A pole vaulter of mass 60 kg runs at 9.0 m/s and clears a bar. a. Find her kinetic energy as she runs. b. If all of it became gravitational potential energy, how high could her center of mass rise? c. Real vaulters lift their center of mass a little more than that. Where can the extra energy come from?

Answer

a.KE = ½ × 60 × 9.0² = 2,430 J.
b.mgh = 2,430 J, so h = 2,430 ÷ (60 × 9.8) = 4.1 m. (Notice the mass cancels: h = v² ÷ 2g = 81 ÷ 19.6.)
c.From her own muscles, energy flowing into the system: she pushes up with her arms on the pole as she rises. The bent pole stores energy as elastic energy and gives it back; its job is to turn her forward kinetic energy into height. (Some is also lost as thermal energy in the pole and her body, so her muscles have to make that up too.)

c.

Using the idea of molecules, explain what happens to the energy of a moving football when it rolls across a grass field and stops. Is energy conserved?

Answer

At the start, all the molecules of the football are moving forward together: this shared motion is the ball’s kinetic energy. As it rolls, the grass bends and the ball squashes a little at the bottom, again and again; rubbing in the grass and the ball’s rubber, and the air pushed aside, pass the energy to the molecules of the grass, the ball and the air. They end up jiggling slightly faster, in random directions: thermal energy. The ball’s orderly motion has become the disorderly motion of many molecules. Yes: energy is conserved. The total is the same; only its form has changed, from kinetic energy of the ball to thermal energy of the surroundings.

d.

A 0.50 kg ball is dropped from 4.0 m and bounces back to 3.0 m. Make an energy model of the bounce: a. choose the system, b. find the energy in each store at the start, just before it hits the floor, and at the top of the bounce, and c. find the energy that became thermal energy. d. How fast was it going just before it hit the floor, and just after it left it?

Answer

a.The ball, the Earth and the floor (and the air around them). Nothing outside does work on them, so no energy flows in.
b.At the start: GPE = 0.50 × 9.8 × 4.0 = 19.6 J, KE = 0. Just before it hits: GPE = 0, KE = 19.6 J. At the top of the bounce: GPE = 0.50 × 9.8 × 3.0 = 14.7 J, KE = 0.
c.19.6 − 14.7 = 4.9 J became thermal energy (and a little sound), 25% of the start.
d.Before: ½ × 0.50 × v² = 19.6, so v = √78.4 = 8.9 m/s. After: ½ × 0.50 × v² = 14.7, so v = √58.8 = 7.7 m/s.

e.

A hydropower plant uses water that falls 100 m through a pipe. a. How much gravitational potential energy does 1 kg of water release? b. If 1,000 kg of water flows through every second and the plant is 85% efficient, what is its power output? c. What happens to the other 15%?

Answer

a.GPE = mgh = 1 × 9.8 × 100 = 980 J.
b.Every second, 1,000 kg releases 1,000 × 980 = 980,000 J. 85% of it is 833,000 J per second, so the power is 833,000 W, about 0.83 MW.
c.It becomes thermal energy: friction in the pipe and the turbine’s bearings, swirling water, heating in the generator’s wires, and sound. (How Much Electricity Can a Small Stream Make? goes further with this.)

f.

Climbing from Bhaktapur (1,401 m) to Nagarkot (2,175 m), a 50 kg student gains gravitational potential energy. a. How much? b. Muscles turn only about a quarter of the food energy they use into work; the rest becomes thermal energy. How much food energy does the climb take, at least? c. A plate of cooked rice (about 300 g) gives about 1.6 MJ of food energy. How many plates is that?

Answer

a.GPE = mgh = 50 × 9.8 × 774 = 379,000 J (379 kJ).
b.379 kJ is a quarter of the energy used, so the food energy is 4 × 379 = 1,520 kJ (about 1.5 MJ). The other three-quarters warms the body, which is why you get hot climbing.
c.About one plate, just for the climbing, on top of what the body uses anyway.

Where to go next

Why Does Heat Never Flow from Cold to Hot? The ball’s energy spreads out into thermal energy, and never gathers itself back to bounce the ball higher. Why energy spreads one way, and why no engine is perfect.
How Much Electricity Can a Small Stream Make? Gravitational potential energy at work: measure a stream’s drop and flow, work out its power, and build a turbine.
How Do Magnets Push Without Touching? More on the energy stored in fields, electric and magnetic.