Work, Energy & Power

Chapter 2 established how a force changes an object's motion. This chapter introduces a genuinely different way of tracking that same physics — one that doesn't need direction at all, only magnitude — and it turns out to be one of the most powerful bookkeeping tools in all of physics: energy.

Work: Force Acting Over a Distance

In physics, work has a precise meaning, narrower than its everyday use. Work is done only when a force causes displacement in the direction of that force:

Here, F is the applied force, d is the displacement, and θ is the angle between the force and the direction of motion. Holding a heavy box perfectly still, no matter how tiring, does zero physics-work, since there is no displacement at all. Carrying that same box horizontally at constant height also does zero work against gravity specifically, since the force (gravity, straight down) is perpendicular (θ = 90°, and cos(90°) = 0) to the horizontal motion.

Worked Example: Pushing a Crate

A warehouse worker pushes a crate 8 metres across a level floor with a constant horizontal force of 45 N. How much work is done?

Because the push and the motion are in the same direction, θ = 0° and cos(0°) = 1, so the equation reduces to the simpler W = Fd for this case.

Kinetic Energy: The Energy of Motion

Any moving object has kinetic energy — energy due to its motion:

This formula has a real, layered history of its own. Between 1676 and 1689, Gottfried Leibniz developed the mathematical concept of vis viva ("living force"), recognising that the quantity mv² stayed conserved in many mechanical systems — a genuine forerunner of kinetic energy, though not yet distinguished from momentum. In the early 1700s, Émilie du Châtelet performed real experiments dropping balls into soft clay, showing that the resulting indentation depth scaled with the square of the impact velocity, not velocity itself — direct experimental evidence for the v² term, distinguishing kinetic energy from momentum (which scales only with v, not v²) for the first time.

Worked Example: A Rolling Bowling Ball

A 6 kg bowling ball rolls down the lane at 4 m/s. What is its kinetic energy?

Potential Energy: Stored Energy of Position

An object raised against gravity stores gravitational potential energy, released as it falls:

Here m is mass, g is the local gravitational acceleration (9.81 m/s² on Earth, as used throughout this course), and h is height above a chosen reference level. Potential energy is always measured relative to some reference point — only changes in height genuinely matter physically.

Conservation of Energy: A Real, Multi-Discoverer Story

The principle that energy is never created or destroyed, only converted between forms, is one of the most important ideas in all of physics — and, like Newton's laws in Chapter 2, it was not the work of one single person in one single moment.

In 1842, German physician Julius Robert von Mayer became the first to state the mechanical equivalence of heat and work in essentially modern form, arguing that heat and mechanical work were both forms of one underlying energy. In 1843, working independently, James Prescott Joule demonstrated the same principle experimentally with his now-famous paddle-wheel apparatus: a falling weight turned a paddle wheel inside an insulated barrel of water, and the resulting rise in water temperature let Joule measure, with real precision (accurate to within about 1/200 of a degree Fahrenheit — exceptional for the 1840s), exactly how much mechanical work corresponded to a given amount of heat. In 1847, Hermann von Helmholtz published a comprehensive theoretical treatment, Über die Erhaltung der Kraft ("On the Conservation of Force"), tying the separate threads together mathematically. By around 1850, engineer William Rankine had coined the now-standard phrase "the law of the conservation of energy" for the unified principle.

Worked Example: A Ball Dropped from a Height

A 2 kg ball is dropped from a height of 5 m. Using conservation of energy (ignoring air resistance), what is its speed the instant before it hits the ground?

At the top, all the ball's energy is potential; at the bottom, all of it has converted to kinetic. Since total energy is conserved:

Notice the mass cancelled out entirely — a heavier ball dropped from the same height reaches the same speed, ignoring air resistance, a real and often surprising consequence of gravitational potential and kinetic energy both scaling with mass in exactly the same way.

Power: The Rate of Doing Work

Power measures how quickly work is done or energy is transferred:

Worked Example: A Stair-Climbing Motor

An electric hoist does 6,000 J of work lifting a load in 12 seconds. What is its power output?

Work, Energy & Power Compared

Hands-On Exercises

Next chapter: Momentum & Collisions — where Chapter 2's own F = dp/dt reappears in its natural home, and conservation of momentum joins conservation of energy as a second, equally powerful bookkeeping tool.