Work–Energy Theorem
When a net force acts on an object and moves it through a distance, it does work on that object. The work–energy theorem tells us precisely what that work accomplishes: it changes the object's kinetic energy. More formally, the net work done on an object equals the change in its kinetic energy. This is one of the most powerful statements in classical mechanics because it connects two seemingly different ideas — force acting over distance, and the motion of an object — into a single, clean equation. It works whether the object speeds up, slows down, or both happen in sequence. Think of pushing a stationary shopping trolley. Every push you give it transfers energy into the trolley, making it move faster. If you then drag your hand against it to slow it down, you do negative work, removing kinetic energy. The theorem accounts for all of this simultaneously. The 'net' in net work is crucial: it means we add up the work done by every force acting on the object — gravity, friction, applied forces, air resistance — and the total tells us exactly how much the kinetic energy changed. Individual forces may add or subtract energy; only their sum matters. The theorem is derived directly from Newton's second law, so it carries the same range of validity: it applies to any object moving at speeds well below the speed of light, in any direction, under any combination of forces. It is not a new law of physics — it is Newton's second law rewritten in the language of energy, which often makes problem-solving far more elegant than tracking forces and accelerations step by step.
- Start with Newton's second law
- Substitute into the definition of work
- Identify the kinetic energy terms
- Interpreting sign and direction
- Why 'net' work matters
Start with Newton's second law
Newton's second law states F_net = ma. For a constant net force acting along the direction of motion, the object accelerates uniformly. We can link this to kinematics using v² = u² + 2as, which gives a = (v² − u²) / (2s).
Substitute into the definition of work
Work done by the net force is W_net = F_net × s = ma × s. Substituting our expression for a: W_net = m × [(v² − u²) / (2s)] × s = ½mv² − ½mu². The displacement s cancels cleanly, revealing the result.
Identify the kinetic energy terms
The quantity ½mv² is the kinetic energy at the end of the motion (final KE), and ½mu² is the kinetic energy at the start (initial KE). So W_net = KE_final − KE_initial = ΔKE. This is the work–energy theorem. It holds even when force varies with position, as can be shown with calculus (integrating F·dx).
Interpreting sign and direction
If W_net is positive, kinetic energy increases — the object speeds up. If W_net is negative, kinetic energy decreases — the object slows down. If W_net is zero (e.g. forces cancel, or force is always perpendicular to motion), speed is unchanged. A satellite in a circular orbit is a perfect example: gravity pulls inward, motion is tangential, no work is done, and speed stays constant.
Why 'net' work matters
Consider a box sliding along a rough floor while you push it. The applied force does positive work; friction does negative work. Only their sum — the net work — equals ΔKE. If friction exactly balances your push, net work is zero and the box moves at constant velocity, consistent with Newton's first law. The theorem is fully consistent with all of Newtonian mechanics.
The Key Equation
- W_{\text{net}} — Net work (Joule (J)): The total work done by all forces acting on the object, taking into account both magnitude and sign.
- \Delta KE — Change in kinetic energy (Joule (J)): The difference between the object's final and initial kinetic energies.
- m — Mass (Kilogram (kg)): The inertial mass of the object, assumed constant in classical mechanics.
- v — Final speed (Metres per second (m s⁻¹)): The speed of the object at the end of the interval being considered.
- u — Initial speed (Metres per second (m s⁻¹)): The speed of the object at the start of the interval being considered.
Worked Example
Problem. A 1200 kg car travelling at 10 m s⁻¹ accelerates along a straight road. The engine exerts a forward force of 4000 N and friction exerts a backward force of 1000 N over a distance of 150 m. Find the car's speed at the end of this distance.
Answer. The car's final speed is approximately 29.2 m s⁻¹.
Origins · 1 minGaspard-Gustave de Coriolis and how this was discovered
Coriolis formalised the concept of 'work' as force times displacement in 1829 and introduced the factor of one-half in the expression for kinetic energy (½mv²), giving the modern form still used today. His 1829 paper 'Du Calcul de l'Effet des Machines' established the mathematical framework from which the work–energy theorem directly follows, making him the key figure in translating Newtonian mechanics into energy-based language. Coriolis is perhaps better known today for the Coriolis effect — the deflection of moving objects on a rotating Earth that causes cyclones to spin in different directions in each hemisphere. Yet his deeper contribution to physics was arguably the rigorous definition of mechanical work and kinetic energy.
Related Concepts
- Kinetic Energy — The work–energy theorem is defined in terms of change in kinetic energy; understanding KE = ½mv² is essential before applying the theorem.
- Work Done by a Force — Net work is the left-hand side of the work–energy theorem; knowing how to calculate W = Fd cos θ is a prerequisite for using the theorem correctly.
- Conservation of Energy — The work–energy theorem is a stepping stone to the broader principle of energy conservation, linking work done against gravity to gains in potential energy.
- Power — Power is the rate at which work is done; understanding the work–energy theorem clarifies what that work is actually achieving in terms of changing an object's motion.