# Gravitational potential energy and kinetic energy

> CIE A-Level Physics · Work, energy and power
> Source: https://www.owlsprep.com/study/cie-9702-u5-gravitational-potential-energy-and-kinetic/

This sub-topic explores how gravitational potential energy and kinetic energy change for objects moving in gravitational fields, and how to apply conservation of mechanical energy to solve common exam problems near Earth's surface.

**Prerequisites:** [Work done by constant forces](https://www.owlsprep.com/study/cie-9702-u5-work-done-by-constant-force/); [Gravitational acceleration near Earth's surface](https://www.owlsprep.com/study/cie-9702-u2-gravitational-acceleration/)

## Learning objectives

- Distinguish between changes in gravitational potential energy (GPE) near Earth's surface and in radial gravitational fields
- Relate changes in GPE to changes in kinetic energy (KE) for objects moving under gravity
- Apply the work-energy principle and conservation of mechanical energy to gravitational problems
- Solve standard exam questions for motion of objects in constant gravitational fields

## Gravitational Potential Energy Near Earth's Surface

**Change in Gravitational Potential Energy (near surface)** — The work done against gravity to move an object between two positions. For constant gravitational acceleration $g$, the change in GPE is $\Delta U = mg\Delta h$, where $m$ = mass of the object and $\Delta h$ = change in vertical height.

*Notation:* \Delta U

*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 $g = 9.81 \text{ m s}^{-2}$.

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

> **tip**
>
> Always check the sign of $\Delta h$: $\Delta h$ is positive when the object moves upwards, giving positive $\Delta U$ (GPE gain). If the object moves downwards, $\Delta h$ is negative, so $\Delta U$ is negative (GPE loss).

## Kinetic Energy and the Work-Energy Principle

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

*Notation:* K

*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: $W_{\text{net}} = \Delta K$. When gravity is the only force doing work, the work done by gravity equals $-\Delta U$, so rearranging gives $\Delta U + \Delta K = 0$, 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?

1. Step 1: Initial kinetic energy is zero because the sprinter starts from rest: $K_1 = 0$
2. Step 2: Calculate final kinetic energy:
3. $$K_2 = \frac{1}{2}mv^2 = \frac{1}{2}(50)(10)^2 = 2500 \text{ J}$$
4. Step 3: By the work-energy principle, net work done equals change in kinetic energy:
5. $$W_{\text{net}} = \Delta K = 2500 - 0 = 2500 \text{ J}$$

**Check your understanding**

Check your understanding of energy changes:

1. 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

   *Why:* Correct! The ball does work against gravity, so GPE increases. It slows down as it rises, so KE decreases.

## Applying Conservation of Energy

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.

> **tip**
>
> Set your zero GPE reference point at the lowest point in the problem to simplify calculations, this makes GPE zero at that point and avoids negative values.

**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 $g = 9.81 \text{ m s}^{-2}$.

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

**Exam command terms**

- **By conservation of energy** — You must state that total energy before equals total energy after, and show you have accounted for all energy forms *(You will get a mark for explicitly stating the principle, even if your calculation is wrong)*

## Common pitfalls

- **Wrong:** Using horizontal distance instead of vertical height to calculate change in GPE
  - Why it fails: GPE depends only on change in vertical position, because gravity acts vertically. Horizontal displacement does not change GPE
  - Correct: Always extract the change in vertical height from the problem, ignore any horizontal distance when calculating ΔGPE
- **Wrong:** Forgetting to check the sign of ΔGPE in conservation equations
  - Why it fails: Wrong signs lead to negative values under the square root for speed, which is impossible
  - Correct: Confirm: object moving up → ΔGPE positive, object moving down → ΔGPE negative, check signs before solving
- **Wrong:** Using the general GPE formula $U = -GMm/r$ for near-surface problems
  - Why it fails: 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: Use $\Delta U = mg\Delta h$ for all near-surface problems, only use the radial GPE formula when asked about orbits or escape velocity
- **Wrong:** Double-counting work done by gravity when using GPE
  - Why it fails: GPE already accounts for work done by gravity, so adding it again as work leads to incorrect total energy
  - Correct: When you use GPE in your total energy calculation, you only add work done by non-gravitational forces like friction or tension

## Cheatsheet

| Quantity | Formula (near Earth surface) | Key Notes |
| --- | --- | --- |
| Change in GPE | $\Delta U = mg\Delta h$ | Δh positive upwards, ΔU positive for GPE gain |
| Kinetic Energy | $K = \frac{1}{2}mv^2$ | 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 | $W_{\text{net}} = \Delta K$ | Net work includes all forces, including gravity |

## 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.

- [Conservation of energy](https://www.owlsprep.com/study/cie-9702-u5-conservation-of-energy/)
- [Power](https://www.owlsprep.com/study/cie-9702-u5-power/)
- [Energy efficiency](https://www.owlsprep.com/study/cie-9702-u5-energy-efficiency/)

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