Gravitational potential energy and kinetic energy

CIE A-Level Physics· 35 min read

1. Gravitational Potential Energy Near Earth's Surface★★☆☆☆⏱ 15 min

📘 Definition

Change in Gravitational Potential Energy (near surface)

\Delta U

The work done against gravity to move an object between two positions. For constant gravitational acceleration $g$ , the change in GPE is $Δ U = m g Δ h$ , where $m$ = mass of the object and $Δ h$ = change in vertical height.

Example:

An object lifted upwards gains positive GPE.

Near Earth's surface, we assume $g$ is constant, so GPE changes linearly with vertical height. We can set the zero GPE reference point anywhere, because we only measure changes in GPE for most problems.

📐 Worked Example

A 2.0 kg textbook is lifted from the floor to a shelf 1.6 m above the floor. Calculate the change in GPE of the book, take $g = 9.81 m s^{- 2}$ .

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Step 1: Identify known values: $m = 2.0 kg$ , $g = 9.81 m s^{- 2}$ , $Δ h = 1.6 m$
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Step 2: Substitute into the formula for change in GPE:
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$\Delta U = mg\Delta h = 2.0 \times 9.81 \times 1.6$
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Step 3: Calculate and round to 2 significant figures:
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$\Delta U = 31 \text{ J (2 s.f.)}$

2. Kinetic Energy and the Work-Energy Principle★★☆☆☆⏱ 15 min

📘 Definition

Kinetic Energy

K

The energy an object has due to its motion, calculated as $K = \frac{1}{2} m v^{2}$ , where $m$ = mass and $v$ = speed of the object.

Example:

A stationary object has zero kinetic energy.

The work-energy principle states that the net work done on an object equals the change in its kinetic energy: $W_{net} = Δ K$ . When gravity is the only force doing work, the work done by gravity equals $- Δ U$ , so rearranging gives $Δ U + Δ K = 0$ , meaning total mechanical energy is constant.

📐 Worked Example

A 50 kg sprinter accelerates from rest to 10 m s⁻¹. What is the net work done on the sprinter?

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Step 1: Initial kinetic energy is zero because the sprinter starts from rest: $K_{1} = 0$
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Step 2: Calculate final kinetic energy:
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$K_2 = \frac{1}{2}mv^2 = \frac{1}{2}(50)(10)^2 = 2500 \text{ J}$
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Step 3: By the work-energy principle, net work done equals change in kinetic energy:
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$W_{\text{net}} = \Delta K = 2500 - 0 = 2500 \text{ J}$

✓ Quick check

Check your understanding of energy changes:

A ball is thrown vertically upwards. Which correctly describes energy changes as the ball rises to maximum height?
- Both GPE and KE increase
- GPE increases, KE decreases
- GPE decreases, KE increases
- Both GPE and KE decrease
Reveal answer
GPE increases, KE decreases —
Correct! The ball does work against gravity, so GPE increases. It slows down as it rises, so KE decreases.

3. Applying Conservation of Energy★★★☆☆⏱ 20 min

When no non-conservative forces (like air resistance or friction) do work, total mechanical energy (sum of GPE and KE) is conserved. This lets us solve problems like motion on curved paths that are difficult with kinematics alone.

📐 Worked Example

A 0.5 kg ball is dropped from rest from a height of 10 m above the ground. Assuming no air resistance, calculate its speed just before impact, take $g = 9.81 m s^{- 2}$ .

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Step 1: Set zero GPE at ground level. Initial energy: ball is at rest, so $K E_{initial} = 0$ , $GP E_{initial} = m g h$
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Step 2: Total initial energy:
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$E_{\text{total, initial}} = 0 + (0.5)(9.81)(10) = 49.05 \text{ J}$
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Step 3: Final energy just before impact: GPE is zero at ground level, so total energy is all kinetic:
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$E_{\text{total, final}} = \frac{1}{2}mv^2 + 0$
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Step 4: Equate total initial and final energy (conservation of energy):
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$mgh = \frac{1}{2}mv^2$
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Step 5: Mass cancels out on both sides, so solve for v:
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$v = \sqrt{2gh} = \sqrt{2 \times 9.81 \times 10} \approx 14 \text{ m s}^{-1} \text{ (2 s.f.)}$

4. Common Pitfalls

Wrong move:

Using horizontal distance instead of vertical height to calculate change in GPE

Why:

GPE depends only on change in vertical position, because gravity acts vertically. Horizontal displacement does not change GPE

Correct move:

Always extract the change in vertical height from the problem, ignore any horizontal distance when calculating ΔGPE

Wrong move:

Forgetting to check the sign of ΔGPE in conservation equations

Why:

Wrong signs lead to negative values under the square root for speed, which is impossible

Correct move:

Confirm: object moving up → ΔGPE positive, object moving down → ΔGPE negative, check signs before solving

Wrong move:

Using the general GPE formula $U = - GM m / r$ for near-surface problems

Why:

This formula uses zero GPE at infinity and is only for large-scale orbital problems. Using it for small height changes near Earth leads to incorrect results

Correct move:

Use $Δ U = m g Δ h$ for all near-surface problems, only use the radial GPE formula when asked about orbits or escape velocity

Wrong move:

Double-counting work done by gravity when using GPE

Why:

GPE already accounts for work done by gravity, so adding it again as work leads to incorrect total energy

Correct move:

When you use GPE in your total energy calculation, you only add work done by non-gravitational forces like friction or tension

5. Quick Reference Cheatsheet

Quantity	Formula (near Earth surface)	Key Notes
Change in GPE	$Δ U = m g Δ h$	Δh positive upwards, ΔU positive for GPE gain
Kinetic Energy	$K = \frac{1}{2} m v^{2}$	Always positive, depends on speed not direction
Conservation of Mechanical Energy	$U_{1} + K_{1} = U_{2} + K_{2}$	Valid when only gravity does work
Work-Energy Principle	$W_{net} = Δ K$	Net work includes all forces, including gravity

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 1
GPE change for moving object
2022 · 2
Conservation of energy for falling object
2021 · 1
KE change for projectile motion

What's Next

Understanding the relationship between GPE and KE is the foundation for all energy topics in A-Level Physics. This concept extends to problems involving non-conservative forces like friction, where energy is dissipated as heat, and to power calculations for objects moving against gravity. Extending GPE to large-scale radial gravitational fields leads to understanding satellite motion and escape velocity, which are common longer questions in CIE A-Level papers. Mastering this sub-topic makes more advanced energy and gravitational problems much easier to solve.