Work-energy theorem — derivation from Newton's second law: Foundations

This page assumes you know nothing but arithmetic and what an arrow-picture means. We build every letter and squiggle in the parent note (parent topic) from the ground up, one at a time, each one earning its place before the next.

0. The stage: a particle on a line

Before any symbol, picture the scene. A single small object — we call it a particle because we ignore its size and treat it as one dot — sits on a straight track. It can be at different places along that track, and it can move.

Figure — Work-energy theorem — derivation from Newton's second law

Everything in this topic happens on this one line first (1-D). Later we add direction-arrows for 3-D.

1. Position $x$ — where the dot is

Picture: a ruler laid along the track. The dot sits above the number $x$ .
Why the topic needs it: work is "force spread over distance". Distance is change in position, so we cannot talk about work without first pinning down where the dot is.

2. A tiny step $d x$ — an almost-nothing piece of distance

Picture: zoom into the ruler until one hair-thin slice remains; that width is $d x$ .
Why the topic needs it: if the force is different at every position, we cannot multiply force by the whole distance at once. Instead we chop the distance into countless slivers $d x$ , handle each tiny piece where the force is nearly constant, then add them all back up. That "add up the slivers" is the integral (§9).

3. Velocity $v$ and the derivative $\frac{d v}{d t}$

Picture: the dot at two nearby instants; $d x$ is the gap it slid, $d t$ the time between snapshots. Their ratio is the speed.

Why the topic needs it: Newton's law (§6) is literally written as $F = m \frac{d v}{d t}$ . And the famous " $v d v$ trick" in the derivation is nothing but juggling these two ratios $\frac{d x}{d t}$ and $\frac{d v}{d t}$ . If you own these symbols, the whole derivation is bookkeeping.

4. Mass $m$ — how hard it is to speed up

Picture: two dots, a heavy one drawn large, a light one small; the same push moves the small one more.
Why the topic needs it: $m$ is the bridge between force and motion in $F = ma$ , and it sits inside kinetic energy $K = \frac{1}{2} m v^{2}$ .

5. Force $F$ — the push or pull

Picture: a horizontal arrow on the dot. Longer arrow = bigger force; arrow pointing back = negative force.

Why the topic needs it: the theorem is about $W_{n e t}$ , built from the net force — not just the force you happen to apply. This single word "net" is the source of the most common mistake in the parent note.

6. Newton's Second Law $F_{n e t} = ma$

Picture: a bigger arrow (force) sitting over a dot that consequently gets a bigger velocity-change arrow, scaled down by heavy mass.
Why the topic needs it: this is the only fundamental input. The Work–Energy Theorem is not a new law of nature — it is $F = ma$ re-organised. See Newton's Second Law.

7. Work $W$ — force added up over distance

Picture: shade the area under a graph of force (up-axis) versus position (across-axis). That shaded area is the work.

Why the topic needs it: $W$ is one half of the theorem. Reading it as "area under the force-vs-position graph" is what makes the variable-force case (a spring) natural — see Work done by a variable force.

8. Kinetic energy $K = \frac{1}{2} m v^{2}$ — motion-energy

Picture: a speedometer wired to an energy-tank; the tank fills faster than the speed rises because of the squared $v$ .
Why it uses $v^{2}$ (a scalar): squaring throws away the sign of velocity, so moving left or right at the same speed gives the same $K$ . Energy has no direction — that is why $K$ is a scalar and always $\geq 0$ . More at Kinetic Energy.

9. The integral sign $\int_{a}^{b}$ — add up all the slivers

Picture: the shaded area of §7, now sliced into thin rectangles of width $d x$ and height $F$ ; the integral sums their areas.
Why the topic needs it: the left side of the theorem, "net work", is this integral. The right side, after the $v d v$ trick, is another integral $\int m v d v$ that evaluates to $\frac{1}{2} m v^{2}$ .

10. Vectors $F, v, d r$ and the dot product $\cdot$

Picture: shine a light straight down onto the displacement arrow; the shadow of the force arrow onto it is the "aligned part" the dot product keeps.
Why the topic needs it: in 3-D, work is $\int F \cdot d r$ . The dot product is the tool that answers "how much of this push actually helps the motion?" — it automatically handles angles and signs, so we never track them by hand. This is why a normal force (always sideways) does zero work.

Prerequisite map

Read it top-down: position gives velocity gives acceleration; mass + force + acceleration give Newton's law; force + distance give work; work summed by the integral, set equal to the change in kinetic energy, is the theorem.

Equipment checklist

Cover the right side and try to answer each before revealing.

What does

d x

mean in one phrase?

A tiny sliver of distance, so small the force is constant across it.

Write velocity as a ratio of tiny changes.

v = \frac{d x}{d t}

What physical quantity is

\frac{d v}{d t}

The acceleration

a

— how fast velocity changes each second.

State Newton's second law with the derivative form.

F_{n e t} = m \frac{d v}{d t}

What does the integral

\int_{x_{i}}^{x_{f}} F d x

compute geometrically?

The area under the force-vs-position graph = the net work.

Why is kinetic energy a scalar (no direction)?

Because

K = \frac{1}{2} m v^{2}

uses speed squared, which erases the sign/direction of velocity.

What does

Δ

Δ K

mean?

Final value minus initial value:

K_{f} - K_{i}

Which force goes into

W_{n e t} = Δ K

The net force — all forces added, not just the applied one.

What does the dot product

F \cdot d r

keep?

Only the part of the force that lines up with the displacement (aligned component).

When is

W = F d

(plain multiply) allowed?

Only when the force is constant and parallel to the motion.

Connections

Newton's Second Law — the single fundamental input every symbol here serves.
Kinetic Energy — where $K = \frac{1}{2} m v^{2}$ is explored fully.
Work done by a variable force — why the integral (§9) is unavoidable.
Power — the time-rate cousin, $P = F \cdot v$ , reusing the dot product from §10.
Impulse–Momentum Theorem — the same $F = ma$ integrated over time instead of distance.
Conservation of Mechanical Energy — what the theorem becomes when forces are conservative.