Chapter 1 of 8

All 8 chapters
  1. 1Calculating energy stores, readunderstood
  2. 2Springs and elastic energy, readunderstood
  3. 3Work done, readunderstood
  4. 4Power, readunderstood
  5. 5Efficiency, readunderstood
  6. 6Specific heat capacity, readunderstood
  7. 7Energy resources, readunderstood
  8. 8Energy at home, readunderstood

S4.9.1 Chapter 1

Calculating energy stores

A car drives at 10 m/s and then at 20 m/s. What happens to its kinetic energy?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

  • You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Calculate kinetic energy and gravitational potential energy
  • Use energy conservation to find the speed of a falling object

The car is the same car, but the faster drive holds more energy. How much more? Energy is counted in joules, written J, so we can work it out exactly. A raised thing holds energy too, and we can count that as well.

Kinetic energy is the energy in the kinetic store of a moving thing. It depends on two things: the mass m in kg and the speed v in m/s.

kinetic energy = ½ × mass × speed². In symbols, Ek = ½ × m × v². The speed is squared. The mass is not.

A bar chart of the kinetic energy of a 2 kg ball. The axis is labelled kinetic energy in J, with a scale from 0 to 16 in steps of 4. The bar for 1 m/s reaches 1. The bar for 2 m/s reaches 4. The bar for 3 m/s reaches 9. The bar for 4 m/s reaches 16.04812161 m/s2 m/s3 m/s4 m/skinetic energy in J
A 2 kg ball. Double the speed, from 1 m/s to 2 m/s, and the energy is four times as big: 1 J becomes 4 J.

Look at the bars. Speeds of 1, 2, 3 and 4 m/s give energies of 1, 4, 9 and 16 J. The energy grows much faster than the speed.

Worked example

A car has a mass of 1 500 kg and moves at 20 m/s. How much kinetic energy does it have?

  1. Ek = ½ × m × v².
  2. v² = 20 × 20 = 400.
  3. Ek = ½ × 1 500 × 400 = 300 000 J = 300 kJ.

Answer: 300 kJ

Three mistakes are common here. First, squaring the mass instead of the speed. Second, leaving out the half. Third, using the wrong units. Mass must be in kg, not g. Speed must be in m/s, not km/h.

To change km/h into m/s, divide by 3.6. A speed of 72 km/h is 72 ÷ 3.6 = 20 m/s.

Gravitational potential energy is the energy in the gravitational potential store of a raised thing. It depends on the mass m, the height h in m, and the gravitational field strength g. The height is measured above the place you choose as zero, such as the floor.

Near the surface of the Earth, g is about 9.8 N/kg. It changes a little from place to place. On the Moon it is about 1.6 N/kg, so the same lift stores less energy. Some books use 10 N/kg to shorten sums. This book uses 9.8 N/kg.

gravitational potential energy = mass × g × height. In symbols, Ep = m × g × h.

A table of the numbers for a lifted bag, with the columns Quantity and Value. The mass m is 3 kg. The height h is 4 m. The gravitational field strength g is 9.8 N/kg.QuantityValuemass m3 kgheight h4 mg9.8 N/kg
The numbers for the bag. Each one has its unit.

Worked example

A bag of 3 kg is lifted 4 m. How much energy does it gain in its gravitational potential store? Use g = 9.8 N/kg.

  1. Ep = m × g × h.
  2. m × g = 3 × 9.8 = 29.4.
  3. Ep = 29.4 × 4 = 117.6 J.

Answer: 117.6 J

Energy is never made or destroyed. When a ball falls, gravitational potential energy changes into kinetic energy. This is the conservation of energy. With no air resistance, the two always add up to the same total.

Draw it first

A falling ball keeps its total

  1. A 2 kg ball is shown at three heights, with a bar beside each. At 5 m the bar says Ep 98 J and Ek 0 J. At 2.5 m it says Ep 49 J and Ek 49 J. At 0 m it says Ep 0 J and Ek 98 J. Every bar is the same length. The ball is labelled 2 kg ball.2 kg ball5 mEp 98 JEk 0 J

    1 At 5 m the ball is still. Ek is 0 J. Ep is 98 J, all of the energy.

  2. A 2 kg ball is shown at three heights, with a bar beside each. At 5 m the bar says Ep 98 J and Ek 0 J. At 2.5 m it says Ep 49 J and Ek 49 J. At 0 m it says Ep 0 J and Ek 98 J. Every bar is the same length. The ball is labelled 2 kg ball.2 kg ball5 mEp 98 JEk 0 J2.5 mEp 49 JEk 49 J

    2 At 2.5 m half the height is gone. Ep is 49 J and Ek is 49 J.

  3. A 2 kg ball is shown at three heights, with a bar beside each. At 5 m the bar says Ep 98 J and Ek 0 J. At 2.5 m it says Ep 49 J and Ek 49 J. At 0 m it says Ep 0 J and Ek 98 J. Every bar is the same length. The ball is labelled 2 kg ball.2 kg ball5 mEp 98 JEk 0 J2.5 mEp 49 JEk 49 J0 mEp 0 JEk 98 J

    3 At 0 m, on the ground, Ep is 0 J and Ek is 98 J. The total is still 98 J.

Two boxes joined by an arrow. The first box says gravitational potential store of the ball. The arrow says as it falls. The second box says kinetic store of the ball.gravitationalpotential store ofthe ballkinetic store of theballas it falls
Energy moves from one store to the other. None of it is lost on the way.

Now use this to find a speed. Just before the ball lands, all the energy is kinetic. So ½ × m × v² = m × g × h. The mass m is on both sides, so it cancels. That leaves v² = 2 × g × h. Then v is the square root of 2 × g × h.

The mass has gone. The speed of a fall does not depend on the mass. This holds when air resistance is too small to matter. In air, a real ball reaches a little less than this speed.

A 0.5 kg ball is shown at three heights, with a bar beside each. At 10 m the bar says Ep 49 J and Ek 0 J. At 5 m it says Ep 24.5 J and Ek 24.5 J. At 0 m it says Ep 0 J and Ek 49 J. The ball is labelled 0.5 kg ball.0.5 kg ball10 mEp 49 JEk 0 J5 mEp 24.5 JEk 24.5 J0 mEp 0 JEk 49 J
A 0.5 kg ball dropped from 10 m. All 49 J ends up as kinetic energy.

Worked example

A ball is dropped from a height of 10 m. Ignore air resistance. How fast is it moving just before it lands? Use g = 9.8 N/kg.

  1. All the gravitational potential energy becomes kinetic energy: ½ × m × v² = m × g × h.
  2. The mass cancels. So v² = 2 × g × h. Here 2 × g = 19.6.
  3. v² = 19.6 × 10 = 196.
  4. v = √196 = 14 m/s.

Answer: 14 m/s

A wrong idea many people have

A heavy object falls faster, so it lands with more speed.

Why it feels right: A heavy stone does hit the ground harder than a feather, and a feather floats down slowly.

The test that breaks it: Drop a small stone and a large stone together from the same height onto soft soil. You hear one thud. In v² = 2 × g × h the mass cancels, so the speed cannot depend on it.

What is true: With no air resistance, everything dropped from the same height lands at the same speed. A heavy object has more kinetic energy at that speed, so it hits harder, but it is not faster. A feather is slowed by the air.

Try it

Do heavy things fall faster?

You need: a small stone, a larger stone, soft ground, such as soil or sand, a friend to listen with you.

Be safe: Drop only onto soft ground and keep your feet well back. Never drop things from a window, a roof or any high place.

  1. Stand on soft ground. Hold the two stones side by side at the height of your chest.
  2. Open both hands at the same moment.
  3. Listen to the landing. Is it one sound or two?
  4. Repeat it three times. Try a potato in place of one stone.

What you will see: You hear one thud each time. Both stones land together, so they speed up in the same way. The larger stone has more kinetic energy when it lands, which is why it digs a deeper hole in the soil.

If you cannot try it, this is what happens.
Two round shapes standing for stones, side by side. The left one is labelled small stone and the right one is labelled large stone. Below them a line is labelled they land together.small stonelarge stonethey land together

Key words

kinetic energy
the energy in the kinetic store of a moving thing, Ek = ½ × m × v².
gravitational potential energy
the energy in the gravitational potential store of a raised thing, Ep = m × g × h.
gravitational field strength
the force of gravity on each kg of mass, written g. Near the Earth it is about 9.8 N/kg.
conservation of energy
energy is never made or destroyed, only moved from one store to another.

Back to your guess

The answer is C: It becomes four times as big.

The speed is squared in the sum. Twice the speed means 2 × 2 = 4 times the kinetic energy.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A cyclist makes her speed five times as big. Her mass stays the same. How many times as big is her kinetic energy?

    Tap your answer.

    1. Yes, that is right. Ek = ½ × m × v². Make the speed five times as big and v² becomes 5 × 5 = 25 times as big. The mass and the half do not change.

    2. Not this one. That would be right if the speed was not squared. Kinetic energy follows the speed squared, so 5 × 5 = 25. Try another.

    3. Not this one. You doubled the 5. The speed is squared, not doubled: 5 × 5 = 25. Try another.

    4. Not this one. Kinetic energy depends on the speed. A faster cyclist has more of it. Try another.

    See the answer

    25 times as big

    Ek = ½ × m × v². Make the speed five times as big and v² becomes 5 × 5 = 25 times as big. The mass and the half do not change.

    • If you chose B: That would be right if the speed was not squared. Kinetic energy follows the speed squared, so 5 × 5 = 25.
    • If you chose C: You doubled the 5. The speed is squared, not doubled: 5 × 5 = 25.
    • If you chose D: Kinetic energy depends on the speed. A faster cyclist has more of it.

  2. A trolley of mass 4 kg moves at 3 m/s. What is its kinetic energy?

    Tap your answer.

    1. Not this one. You used the speed, 3, where the sum needs the speed squared, 9. Square only the speed. Try another.

    2. Yes, that is right. Ek = ½ × m × v² = ½ × 4 × 9 = 18 J.

    3. Not this one. You squared the mass, 4 × 4 = 16, instead of the speed. Square only the speed. Try another.

    4. Not this one. You left out the half. 4 × 9 = 36 is twice the kinetic energy. Try another.

    See the answer

    18 J

    Ek = ½ × m × v² = ½ × 4 × 9 = 18 J.

    • If you chose A: You used the speed, 3, where the sum needs the speed squared, 9. Square only the speed.
    • If you chose C: You squared the mass, 4 × 4 = 16, instead of the speed. Square only the speed.
    • If you chose D: You left out the half. 4 × 9 = 36 is twice the kinetic energy.

  3. A box of 5 kg is lifted onto a shelf 3 m high. Use g = 9.8 N/kg. How much gravitational potential energy does it gain?

    Tap your answer.

    1. Not this one. You multiplied mass by height and forgot g. Ep = m × g × h uses all three. Go to S3.7.4. Try another.

    2. Not this one. You stopped at m × g, the weight in newtons. The height still has to be multiplied in. Try another.

    3. Yes, that is right. Ep = m × g × h = 5 × 9.8 × 3 = 147 J.

    4. Not this one. You put in a half. The half belongs to kinetic energy, not to Ep. Try another.

    See the answer

    147 J

    Ep = m × g × h = 5 × 9.8 × 3 = 147 J.

    • If you chose A: You multiplied mass by height and forgot g. Ep = m × g × h uses all three. Go to S3.7.4.
    • If you chose B: You stopped at m × g, the weight in newtons. The height still has to be multiplied in.
    • If you chose D: You put in a half. The half belongs to kinetic energy, not to Ep.

  4. Two balls are dropped from the same height. Ignore air resistance. Just before landing, which statement is true?

    Two balls side by side at the same height. The left ball is labelled 1 kg ball and the right ball is labelled 5 kg ball.1 kg ball5 kg ball
    Tap your answer.

    1. Not this one. In v² = 2 × g × h the mass cancels. A heavier ball does not gain more speed. Go to S2.5.5. Try another.

    2. Not this one. Lighter does not mean faster. With no air resistance both balls speed up in the same way. Go to S2.5.5. Try another.

    3. Not this one. They have the same speed, but the 5 kg ball has five times the mass. So it has five times the kinetic energy. Try another.

    4. Yes, that is right. The speed after a fall does not depend on the mass. Kinetic energy does, because Ek = ½ × m × v².

    See the answer

    Both move at the same speed, and the heavier ball has more kinetic energy.

    The speed after a fall does not depend on the mass. Kinetic energy does, because Ek = ½ × m × v².

    • If you chose A: In v² = 2 × g × h the mass cancels. A heavier ball does not gain more speed. Go to S2.5.5.
    • If you chose B: Lighter does not mean faster. With no air resistance both balls speed up in the same way. Go to S2.5.5.
    • If you chose C: They have the same speed, but the 5 kg ball has five times the mass. So it has five times the kinetic energy.

  5. Which sum gives the speed v of a ball after it falls a height h? Ignore air resistance.

    Tap your answer.

    1. Not this one. That has no square root. From ½ × m × v² = m × g × h we get v² = 2 × g × h, and v is the root of that. Try another.

    2. Yes, that is right. All the gravitational potential energy becomes kinetic energy. ½ × m × v² = m × g × h. Cancel m, double both sides, and take the square root.

    3. Not this one. A factor of 2 and a square root are both missing. Start from ½ × m × v² = m × g × h. Try another.

    4. Not this one. The 2 is missing. Move the half across: v² = 2 × g × h. Try another.

    See the answer

    v is the square root of twice g times h

    All the gravitational potential energy becomes kinetic energy. ½ × m × v² = m × g × h. Cancel m, double both sides, and take the square root.

    • If you chose A: That has no square root. From ½ × m × v² = m × g × h we get v² = 2 × g × h, and v is the root of that.
    • If you chose C: A factor of 2 and a square root are both missing. Start from ½ × m × v² = m × g × h.
    • If you chose D: The 2 is missing. Move the half across: v² = 2 × g × h.

  6. A car of mass 1 000 kg moves at 72 km/h. What must you do before you use Ek = ½ × m × v²?

    Tap your answer.

    1. Yes, that is right. Joules come out only when the mass is in kg and the speed is in m/s. Divide km/h by 3.6.

    2. Not this one. The sum needs the mass in kg. Grams would make the answer a thousand times too big. Go to M2.6.4. Try another.

    3. Not this one. The joule needs the speed in m/s. With km/h the answer is wrong by a large factor. Try another.

    4. Not this one. To change km/h into m/s you divide by 3.6. Multiplying gives a speed that is far too big. Try another.

    See the answer

    Change the speed from km/h to m/s.

    Joules come out only when the mass is in kg and the speed is in m/s. Divide km/h by 3.6.

    • If you chose B: The sum needs the mass in kg. Grams would make the answer a thousand times too big. Go to M2.6.4.
    • If you chose C: The joule needs the speed in m/s. With km/h the answer is wrong by a large factor.
    • If you chose D: To change km/h into m/s you divide by 3.6. Multiplying gives a speed that is far too big.

  7. A ball of mass 0.5 kg moves at 10 m/s. What is its kinetic energy in joules?

    See the answer

    25 J

    Ek = ½ × 0.5 × 10 × 10 = ½ × 0.5 × 100 = 25 J.

  8. A stone is dropped from a height of 22.5 m. Ignore air resistance. How fast is it moving just before it lands? Use g = 9.8 N/kg.

    See the answer

    21 m/s

    v² = 2 × g × h = 2 × 9.8 × 22.5 = 441, and v = √441 = 21 m/s.

  9. A mass of 2 kg is raised 3 m. Use g = 9.8 N/kg. How much gravitational potential energy does it gain, in joules?

    See the answer

    58.8 J

    Ep = m × g × h = 2 × 9.8 × 3 = 58.8 J.

Keep it

Copy this into your exercise book.

Ek = ½ × m × v². Ep = m × g × h, with g = 9.8 N/kg. A falling thing turns Ep into Ek. Ignoring air, v² = 2 × g × h, and the mass cancels.

Draw: A ball at three heights, with a bar beside each. Every bar is the same length, split into Ep and Ek.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (758 KB). It opens on any phone.