Gravitational Potential and Kinetic Energy

What this lesson covers in the specification

Incandio is aligned to this specification. It is not published by, endorsed by or affiliated with Pearson, and it reproduces none of Pearson's wording — the statement numbers are given so you can check every lesson against your own copy.

• 4.13 — Use GPE = m × g × h
• 4.14 — Use KE = ½ × m × v²
• 4.15 — How conservation of energy links GPE, KE and work

1 · Understand it

No exam language yet. The only question this section answers is: do I actually understand what is happening?

Lift something and you do work against gravity, and that work is now stored: the object can do it back on the way down. The energy in the GRAVITATIONAL POTENTIAL store is GPE = m × g × h — mass, gravitational field strength, and the height it has been raised. It follows directly from the previous page, because the force needed to lift at a steady speed is the weight, mg, and the work done is that force times the height.

Anything moving has energy in its KINETIC store, and the equation is KE = ½ × m × v². The mass appears as you would expect: twice the mass, twice the energy. The speed does not. It appears SQUARED, so twice the speed gives FOUR times the kinetic energy, and three times the speed gives nine times.

That square is not a detail of the algebra; it is the single most consequential fact in this topic, and it took a serious argument to establish. Through the early eighteenth century physicists disagreed about which quantity measured the 'force' of a moving body — mass times velocity, or mass times velocity squared. The dispute was settled partly by experiment: brass balls dropped into soft clay sank to a depth that went as the square of the speed, not in proportion to it. Both quantities turned out to be real and useful — the first is momentum and the second is kinetic energy — but they are different, and confusing them is still one of the commonest errors in mechanics.

Think of it like the difference between a nuisance and a catastrophe

A car at 20 mph and the same car at 40 mph do not differ by a factor of two in any way that matters to whatever it hits. The kinetic energy is four times greater, so four times as much work has to be done to stop it, and if that work is done by a wall in a fraction of a second the forces are four times larger. Every road-safety figure that seems disproportionate — the survival rate for a pedestrian struck at 20 mph against 40 mph, the length of a motorway crash barrier, the depth of a crumple zone — comes from that square. Intuition works in proportions, and energy does not.

The two equations, and the conversion between them

A 0.50 kg ball is dropped from a height of 1.8 m. Take g as 10 N/kg. Find its GPE before it is dropped, and its speed just before it lands, assuming no air resistance.

1. GPE = m × g × h = 0.50 × 10 × 1.8 = 9.0 J.
2. By conservation of energy, all of that GPE has become KE just before impact, so KE = 9.0 J.
3. KE = ½ × m × v², so 9.0 = ½ × 0.50 × v² = 0.25 × v².
4. v² = 9.0 ÷ 0.25 = 36.
5. v = √36 = 6.0 m/s.

Answer: 9.0 J of GPE, arriving at 6.0 m/s.

Statement 4.15 — how the three quantities link, and why it is worth having

1. WORK DONE lifting an object equals the GPE it gains: both are m × g × h.
2. As it falls, GPE is transferred to the KINETIC store, so at any point GPE lost = KE gained.
3. At the bottom, if air resistance is ignored, ALL the GPE has become KE — which is the step that lets you find a landing speed without knowing anything about the time taken.
4. Going the other way, a moving object rising up a slope converts KE back into GPE, and stops when it has all been converted.
5. In reality some energy is transferred to the thermal store by air resistance and friction, so the KE at the bottom is slightly LESS than the GPE at the top. The total is still conserved; some of it has just gone somewhere unhelpful.

That last point is worth taking seriously rather than treating as a disclaimer. Questions routinely say 'assume no air resistance' or 'assume the slope is frictionless', and those phrases are not politeness — they are telling you that you may equate the two energies exactly. When a question does NOT say it, you should expect the numbers not to match and be ready to say where the difference went.

One practical note about the arithmetic of KE. The order of operations matters: SQUARE THE SPEED FIRST, then multiply by the mass, then halve. Squaring after halving, or forgetting the half altogether, are the two commonest slips. And when finding a speed from a kinetic energy, remember the last step is a square root — an answer that comes out enormous has usually stopped one operation early.

2 · Grade 9 Notes

A different job from the section above. You have already understood it; this is the precise set of things to LEARN — definitions to reproduce word for word, processes in order, equations with units, and the answers that score full marks.

GPE = m × g × h

GPE is gravitational potential energy in joules (J), m is mass in kilograms (kg), g is gravitational field strength in newtons per kilogram (N/kg), h is the height raised in metres (m)

Units: J, kg, N/kg, m. On Earth take g as 10 N/kg unless told otherwise. Rearranged: h = GPE ÷ (m × g).

KE = ½ × m × v²

KE is kinetic energy in joules (J), m is mass in kilograms (kg), v is speed in metres per second (m/s)

Units: J, kg, m/s. SQUARE THE SPEED FIRST. Rearranged: v = √(2 × KE ÷ m).

Learn this definition · Gravitational potential energy

The energy in an object's gravitational potential store because of its height above a reference level, equal to its mass multiplied by the gravitational field strength and by the vertical height raised.

Learn this definition · Kinetic energy

The energy in an object's kinetic store because of its motion, equal to half its mass multiplied by the square of its speed.

What the square in the kinetic energy equation means

• DOUBLE the speed → FOUR times the kinetic energy; TRIPLE it → NINE times
• So four times as much work is needed to stop a car going twice as fast
• This is why BRAKING DISTANCE roughly quadruples when the speed doubles
• Mass behaves normally: double the mass, double the kinetic energy
• Momentum (m × v) and kinetic energy (½mv²) are DIFFERENT quantities — do not confuse them

Statement 4.15 — finding a landing speed from a height

1. Calculate the GPE at the top: m × g × h.
2. State that, ignoring air resistance, all of it becomes KE at the bottom.
3. Set ½ × m × v² equal to that value.
4. Rearrange to v² = 2 × KE ÷ m.
5. Take the square root to get v — and notice the mass cancels, so all objects land at the same speed.

Momentum against kinetic energy — both involve mass and speed

A. MOMENTUM = m × v. A vector, in kg m/s, conserved in every collision. Doubling the speed doubles it.

B. KINETIC ENERGY = ½ × m × v². A scalar, in joules, and NOT conserved in a collision where things deform. Doubling the speed quadruples it.

Model answer [3 marks]

A 60 kg student climbs stairs of vertical height 4.5 m. Take g as 10 N/kg. Calculate the gain in gravitational potential energy. [3]

Using GPE = m × g × h, the gain is 60 × 10 × 4.5 = 2700 J.

Model answer [4 marks]

A 1200 kg car travels at 20 m/s. Calculate its kinetic energy, and state what happens to that value if the speed doubles. [4]

Using KE = ½ × m × v², the kinetic energy is ½ × 1200 × 20² = ½ × 1200 × 400 = 240 000 J. If the speed doubles to 40 m/s, the kinetic energy becomes four times greater, because it depends on the square of the speed, so it would be 960 000 J.

Not this: A car going twice as fast has twice as much kinetic energy.

This: It has FOUR times as much, because the speed is squared. This is why stopping distances rise so steeply with speed, and it is the single most important consequence of the equation.

Mark-losing trap. SQUARE THE SPEED FIRST, then multiply by the mass, then halve. Order matters.

Mark-losing trap. Finding a speed from KE ends with a SQUARE ROOT. Do not stop at v².

Mark-losing trap. GPE uses the VERTICAL height raised, not the distance along a slope.

Mark-losing trap. Momentum is mv; kinetic energy is ½mv². Different quantities, different units.

3 · Prove it — the five questions

The five questions climb Grade 6 → Grade 7 → Grade 8 → Grade 9 → Grade 9 challenge, and are marked inside Incandio on your own device, by rule, with an authored diagnosis of the mistake you actually made. The mark schemes stay in the app so that the practice is worth doing; the questions themselves are here.

1. Grade 6 · Calculate [2 marks] — A 4.0 kg object is lifted through a height of 3.0 m. Take g as 10 N/kg. Calculate its gain in gravitational potential energy.
2. Grade 7 · State [2 marks] — A car doubles its speed. What happens to its kinetic energy?
3. Grade 8 · Calculate [3 marks] — A 1500 kg car travels at 12 m/s. Calculate its kinetic energy.
4. Grade 9 · Calculate [5 marks] — A 0.20 kg ball is dropped from a height of 5.0 m. Take g as 10 N/kg and ignore air resistance. Calculate its speed just before it hits the ground.
5. 9+ · Analyse [6 marks] — A road-safety campaign states that a car at 40 mph carries four times the kinetic energy of the same car at 20 mph. A driver replies: 'That cannot be right — 40 is only twice 20.' Select every statement that belongs in a full-mark analysis.

The people behind this science

Two ways into the same idea — the one who argued that it goes as the square, and was right, and the one who showed a falling body ends up with what it started with. Inside Incandio each of them answers knowing exactly which lesson you have just finished.

Émilie du Châtelet — the one who argued that it goes as the square, and was right

The ½mv² on this page is the settlement of an argument she was central to. Through the 1730s and 1740s physicists disputed what quantity measures the force of a moving body: the Newtonians said mass times velocity, and du Châtelet, following Leibniz, argued for a quantity going as v². She marshalled 's Gravesande's experiments — brass balls dropped into soft clay, sinking to a depth that went as the square of the speed rather than in proportion to it — and argued the case in print against the leading French physicists. She was right, and the quantity is kinetic energy.

• “What did the brass balls dropped into clay actually show?”
• “Why were the Newtonians so sure it went simply as the speed?”
• “How do you win an argument against the leading physicists of your country?”
• “Are both quantities real, or was one side simply wrong?”
• “What did it take to do this work when the academies were closed to you?”

Galileo Galilei — the one who showed a falling body ends up with what it started with

Galileo performed an experiment that is conservation of energy in everything but name. He released a pendulum from a measured height and placed a peg in the path of the string, so that the bob swung round a much tighter arc on the far side — and found it always rose to the same height as it had been released from, whatever the peg did to the shape of its path. Something was being conserved that depended on height alone. He also established that falling bodies of different masses reach the ground together, which is the mass cancelling out in exactly the way this page describes.

• “What did the pendulum and the peg show you?”
• “Why did the bob always rise back to the same height?”
• “Why do heavy and light objects fall at the same rate?”
• “What did you think was being preserved as it swung?”
• “How did you measure time accurately enough to study falling?”

Carry on through the course

• Previous lesson: Work Done
• Next lesson: Power
• Energy Stores and Transfers — Where energy sits, how it moves, and why it is never used up however much it feels like it.
• Efficiency, Sankey Diagrams and Insulation — Working out what share of the energy did the job you wanted — and why the answer is never all of it.
• Conduction, Convection and Radiation — Three ways energy moves from hot to cold — and the one question that tells them apart.
• Work Done — The precise meaning of a word everybody thinks they already know — and why holding something still is not work at all.
• Power — Not how much energy, but how fast — and why that is the number people actually buy.
• All of Physics · Physics key terms · required practicals · equations · Incandio Science · The 4PH1 specification