Historical Context & Motivation
Long before physicists had formal equations, engineers and inventors wrestled with a practical question: how much effort does it take to move something, and how fast can a machine do it? The concepts of work, energy, and power grew out of centuries of experimentation with levers, waterwheels, and steam engines. Understanding this history helps you see why these ideas are so central to physics—and why they remain essential for solving real problems today.
These milestones reveal a recurring theme: physicists kept refining the idea that forces acting over distances transfer something measurable—energy—and that the rate of that transfer matters just as much as the total amount. In the IB Physics A.3 topic, you will use these ideas to solve problems about objects speeding up, slowing down, rising, falling, and everything in between.
Core Principles & Definitions
Before diving into calculations, you need a solid grip on the three big ideas and how they connect. Each one builds on the previous concept, forming a chain from force to energy to the rate at which energy is transferred.
Work (W)
Kinetic Energy (Eₖ)
Gravitational Potential Energy (Eₚ)
Conservation of Energy
Power (P)
Visual Explanation — Work and Energy Transfers
This diagram captures the most important detail of the work equation: only the component of force along the displacement does work. If you push a suitcase across the floor at an angle, the horizontal part of your push moves the suitcase, while the vertical part simply presses it into the ground. When the force is perpendicular to the displacement (θ = 90°), cos 90° = 0 and no work is done at all. This is why the normal force and gravity do no work on a box sliding on a level surface—they act vertically while the box moves horizontally.
Mathematical Framework
The equations in this section are the toolkit you will use for almost every problem in A.3. Make sure you understand what each variable represents and the conditions under which each equation applies.
Energy Transfers — Bar Chart Model
One of the most powerful tools for solving energy problems is the energy bar chart. It lets you visualize how energy is distributed at different moments during a process, making it much harder to forget a term or mix up a sign. The diagram below shows a ball being thrown upward: at the bottom it has maximum kinetic energy, and at the top that energy has been fully converted to gravitational potential energy.
When friction or air resistance is present, the total bar height decreases from one snapshot to the next because some mechanical energy has been transferred to thermal energy in the surroundings. In that case, you would add a third bar labeled Ethermal that grows as the other two shrink. Drawing these bar charts before writing equations is a strategy recommended by IB examiners, because it forces you to account for every energy store involved.
Worked Example — Roller Coaster with Friction
A 600 kg roller coaster car starts from rest at the top of a 35 m hill. It reaches the bottom of the hill with a speed of 22 m s⁻¹. Determine the work done by friction and the average friction force if the track length from top to bottom is 80 m.
Strengths and Limitations of the Energy Approach
You have two main tools for solving mechanics problems: Newton's second law (forces and acceleration) and the energy approach. Each has its strengths and weaknesses, and recognizing when to use which can save you significant time on exams.
| Feature | Energy Approach | Newton's Second Law |
|---|---|---|
| Best for | Finding speeds, heights, or distances when the path doesn't matter | Finding forces, accelerations, or analysing motion at a specific instant |
| Path dependence | Works regardless of path shape (for conservative forces); only start and end states matter | Requires knowledge of the path to resolve forces along it |
| Vector vs. scalar | Scalar quantities—no need to break into components along x and y | Vector equations—must resolve forces into components |
| Time information | Does not directly give time; must combine with kinematics if time is needed | Can find time through kinematic equations once acceleration is known |
| Friction handling | Friction appears as negative work or energy lost to thermal energy | Friction is treated as a force opposing motion in the free-body diagram |
Connection to Advanced Theory
The work–energy ideas you learn in A.3 are not just useful for toy problems—they form the backbone of more advanced physics. The table below shows how the same concepts extend into higher-level topics you may encounter in HL Physics, university physics, or engineering courses.
| A.3 Concept | Advanced Extension |
|---|---|
| W = Fd cos θ (constant force) | W = ∫ F · ds — work is the integral of force over a path, allowing variable forces to be handled |
| Eₚ = mgΔh (near Earth's surface) | Eₚ = −GMm/r — the full gravitational potential energy formula for any distance from a planet, used in orbital mechanics |
| Eₖ = ½mv² | Relativistic kinetic energy: Eₖ = (γ − 1)mc², needed when objects approach the speed of light |
| Conservation of mechanical energy | First law of thermodynamics (ΔU = Q − W) — energy conservation including heat transfer and internal energy |
| Power P = W/t | Instantaneous power P = dW/dt — a calculus-based formulation that handles time-varying work |
You don't need to memorize these advanced formulas now, but it's worth knowing that every equation in A.3 is a simplified version of a deeper, more general principle. When you master the simplified forms, you are building the intuition that makes the advanced versions feel natural later. In the IB HL course, you will already encounter the gravitational potential formula Ep = −GMm/r in the astrophysics and gravitation topics.
Practice Problems
Lesson Summary
Work is the transfer of energy when a force acts over a displacement, calculated as W = Fd cos θ. Only the component of force parallel to the displacement contributes. Kinetic energy (½mv²) depends on speed squared, while gravitational potential energy (mgΔh) depends linearly on height. The work–energy theorem states that the net work on an object equals its change in kinetic energy, providing a powerful shortcut for many mechanics problems.
Conservation of energy means the total energy of an isolated system is constant—it only changes form. When friction is present, mechanical energy decreases as thermal energy increases. Power (P = W/t or P = Fv) measures the rate of energy transfer and is crucial for understanding machines and engines. Use energy bar charts to visualize transfers, choose the energy approach when the path doesn't matter and time isn't needed, and always check that friction does negative work.