Conservation of mechanical energy — derivation: Foundations

Before we can prove that total stays frozen, we must know exactly what every scribble in the parent note means. Below, each symbol is built from nothing: plain words → a picture → why the topic needs it. Read top to bottom; every item leans on the one above it. The little "§" sign just means "the numbered section on this page" — e.g. "§8" points you to section 8 below.

1. Mass — $m$

Picture: a solid block sitting on a table, tagged with " $m$ ". Why we need it: every energy formula ( $\frac{1}{2} m v^{2}$ , $m g h$ ) is scaled by $m$ — double the mass, double the energy at the same speed or height.

2. Time — $t$

Picture: a stopwatch ticking beside the moving block. Why we need it: velocity and acceleration are both "per second," so every rate of change is measured against $t$ . A tiny tick of the clock is written $d t$ (a sliver of time, §3).

3. Position and displacement — $x$ , $h$ , $d x$

Picture: a number line; a dot at $x$ , an arrow of length $d x$ nudging it right — see the figure below. Why we need it: work is force acting over a distance, so we must be able to talk about "a little bit of movement" $d x$ and add up all the little bits.

The figure below fixes these three ideas — position $x$ , a sliver $d x$ , and how a sliver of distance over a sliver of time gives velocity — in one picture. Notice the yellow step $d x$ nudging the blue dot rightward, and the green label reading off $v = d x / d t$ .

Figure — Conservation of mechanical energy — derivation

4. Velocity and acceleration — $\frac{d x}{d t}$ , $\frac{d v}{d t}$

Picture: the same dot on the number line, now with a speedometer needle attached.

Why we need it: the whole derivation starts from Newton's law $F = m \frac{d v}{d t}$ , so we must be fluent in "rate of change" before line one. The fraction $\frac{d v}{d t}$ is not two numbers divided — it is one idea: steepness of the speed-vs-time graph.

5. Force — $F$ , $F_{n e t}$

Picture: a block with several arrows on it (gravity down, hand pushing right); their sum is one bold $F_{n e t}$ arrow. Note on notation: whenever a quantity has a direction (force $F$ , displacement $d$ , acceleration $a$ , velocity $v$ ) we may put an arrow on it. Along a single straight line we often drop the arrow and keep just a sign ( $+$ or $-$ ) to record the direction — that is all the "direction" a 1-D line has. Why we need it: Newton's second law and work both start from force. We link which kind of force is acting to decide whether energy is conserved.

6. Newton's second law — $F_{n e t} = m a$

7. Work — $W = F \cdot d$

Picture: a force arrow and a displacement arrow. Work counts only the part of the force lined up with the motion.

Force along motion → positive work (speeds up).
Force against motion → negative work (slows down).
Force sideways (90°) → zero work, no matter how strong.

8. Kinetic energy and the Work–energy theorem — $K = \frac{1}{2} m v^{2}$

Picture: the moving block, a glowing halo whose brightness grows with speed.

Why we need it: it is one of the two pillars the whole proof stands on. It converts the abstract "work" (§7) into a concrete change in the motion-pocket $K$ . Full statement and derivation live in Work-energy theorem.

9. Integration — $\int$

Picture: many thin rectangles under a curve, their areas summed into one shaded region. Why we need it: work over a real distance, and potential energy, are both sums of infinitely many tiny contributions. Without the integral we could only handle one sliver at a time.

10. Potential energy — $U$ — and its link to force

Picture: two "batteries" — one is a ball perched high on a shelf, the other a compressed coil.

$g$ = gravitational field strength, about $9.8 m/s^{2}$ near Earth — how strongly the ground pulls each kilogram. See Gravitational potential energy.
$k$ = spring constant — how stiff a spring is; big $k$ = hard to squash. See Spring potential energy (Hooke's law).

11. Change symbol — $Δ$

Why we need it: the entire conclusion is written with it — $Δ (K + U) = 0$ literally says "the total did not change from start to finish."

12. Non-conservative force (e.g. friction)

Picture: a block sliding on rough ground, tiny heat-squiggles rising behind it. Why we need it: the conservation law only holds when such forces do no net work. Naming this "sticky" energy-thief precisely is what lets us state the exact condition for the proof.

13. Mechanical energy — $E = K + U$

Picture: two pockets whose contents shift back and forth, but a scale weighing both together never moves.

How the foundations feed the topic

Read it top-down: time, mass and rate-of-change build Newton's law; force, displacement and integration build work; work gives the work–energy theorem; force + integration also build potential energy through $F = - d U / d x$ ; and the two "change" boxes collide (with friction absent) to give conservation.

Equipment checklist

Test yourself — cover the right side.

What does

m

measure, in plain words?

How much stuff an object has; its resistance to being moved or stopped (kg).

What is

t

and what is

d t

t

is the clock reading in seconds;

d t

is an infinitesimal sliver of time.

What does the little

d

d x

mean?

An infinitesimally small chunk — a sliver — of that quantity.

What is

\frac{d v}{d t}

in words?

How fast the velocity itself is changing, i.e. acceleration.

State Newton's second law.

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

— net force equals mass times acceleration.

When may we drop the arrow on a vector?

In one dimension, where a

+

-

sign records the only two possible directions.

What does the dot in

F \cdot d

do?

Keeps only the part of the force pointing along the motion; perpendicular forces give zero.

State the work–energy theorem.

Net work equals the change in kinetic energy,

W_{n e t} = Δ K

Write kinetic energy and say why

v

is squared.

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

; doubling speed quadruples the energy.

What does

\int F d x

compute?

The total work by adding up every

F d x

sliver along the path.

State the force–potential-energy link and its sign rule.

F = - \frac{d U}{d x}

, equivalently

W_{co n s} = - Δ U

: positive work by the force means

U

drops.

Why can we choose where

U = 0

Only

Δ U

(the slope of

U

) matters physically; adding a constant changes no force or prediction.

Give the two potential energies used here.

U_{g r a v} = m g h

(zero at chosen floor) and

U_{s p r in g} = \frac{1}{2} k x^{2}

(zero at natural length).

What does

Δ

mean?

Final value minus initial value (positive = increase, negative = decrease).

What is a non-conservative force?

One whose work depends on the path (e.g. friction), draining

K + U

into heat.

What is mechanical energy

E

E = K + U

, the sum of motion and stored energy.

Foundations — Conservation of mechanical energy — derivation

1. Mass — $m$

2. Time — $t$

3. Position and displacement — $x$ , $h$ , $d x$

4. Velocity and acceleration — $\frac{d x}{d t}$ , $\frac{d v}{d t}$

5. Force — $F$ , $F_{n e t}$

6. Newton's second law — $F_{n e t} = m a$

7. Work — $W = F \cdot d$

8. Kinetic energy and the Work–energy theorem — $K = \frac{1}{2} m v^{2}$

9. Integration — $\int$

10. Potential energy — $U$ — and its link to force

11. Change symbol — $Δ$

12. Non-conservative force (e.g. friction)

13. Mechanical energy — $E = K + U$

How the foundations feed the topic

Equipment checklist

Connections