Conservation of mechanical energy — derivation: Question bank

Each line below is Prompt ::: Answer — the answer always carries the why, never a bare yes/no.

Symbol glossary — read this first

Before you argue about a law, you must know exactly what each letter stands for. Every trap below reuses these; keep this table in view.

Definition Every symbol used on this page

$m$ — mass of the object (kilograms). How much "stuff" resists being sped up.
$v$ — speed of the object (metres per second). How fast it moves right now.
$h$ — height above a chosen reference level (metres). "How high up."
$x$ — stretch or compression of a spring away from its relaxed length (metres).
$g$ — gravitational field strength near Earth ( $\approx 9.8 m/s^{2}$ ). How hard gravity pulls per kilogram.
$k$ — spring constant (newtons per metre). How stiff the spring is. See Spring potential energy (Hooke's law).
$F$ — a force (newtons): a push or pull, with a direction.
$d$ — the displacement: the straight arrow from where the object started to where it ended.
$K$ — kinetic energy $= \frac{1}{2} m v^{2}$ : energy stored in motion.
$U$ — potential energy: energy stored in position (e.g. gravitational $m g h$ , spring $\frac{1}{2} k x^{2}$ ). See Potential energy.
$Δ$ (delta) — "change in": $Δ U = U_{after} - U_{before}$ .

One picture — energy as two swapping tanks

Figure — Conservation of mechanical energy — derivation

Recall Where does

$F d x = m v d v$ come from? (the gap the derivation fills) Newton's 2nd law says $F = m \frac{d v}{d t}$ . Multiply both sides by a tiny displacement $d x$ : $F d x = m \frac{d v}{d t} d x$ . Now use $d x = v d t$ (displacement = speed × time): the $d t$ cancels, giving $F d x = m v d v$ . Integrating the left from start to finish gives total work $W$ ; integrating the right, $\int_{v_{1}}^{v_{2}} m v d v = \frac{1}{2} m v_{2}^{2} - \frac{1}{2} m v_{1}^{2} = Δ K$ . That is the work–energy theorem $W_{net} = Δ K$ .

True or false — justify

The total energy of the universe is always conserved, therefore mechanical energy

K + U

is always conserved.

False — the total is conserved, but friction and other non-conservative forces move energy into heat/sound, which lives outside

K + U

. See Energy lost to friction.

If a force is large, it must do a large amount of work.

False — work is

W = F \cdot d

, so a force perpendicular to the motion does zero work no matter how huge it is (e.g. normal force, string tension).

For a ball thrown straight up, its mechanical energy at the top is less than at launch because it has stopped moving.

False — kinetic energy

K

is zero at the top, but the lost KE became gravitational PE

U

; the sum

K + U

is unchanged (ignoring air).

Potential energy is a property that an object "has" all by itself.

False — PE belongs to a system (object + the field/spring producing the conservative force); a lone mass with no field has no PE. See Potential energy.

You can add the work done by gravity and the term

m g h

into one energy equation.

False — that double-counts.

m g h

is the bookkeeping for gravity's work; use force or PE, never both.

Only conservative forces are allowed to be present for mechanical energy to be conserved.

False — non-conservative forces may be present as long as they do no net work (like the perpendicular normal force on a frictionless slope).

A satellite in a circular orbit conserves mechanical energy even though gravity constantly pulls on it.

True — gravity is conservative, and no non-conservative force acts, so

K + U

stays constant (speed and radius both constant here).

Doubling an object's speed doubles its kinetic energy.

False —

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

depends on

v^{2}

, so doubling speed quadruples KE.

The zero level of gravitational PE must be placed at the ground.

False — the zero of

U

is a free choice; only differences

Δ U

enter physics, so any reference height gives the same speeds.

Spot the error

"A block slides down a rough ramp; I'll use

m g h = \frac{1}{2} m v^{2}

to find the bottom speed."

Error — friction is non-conservative and does negative work, so you must write

m g h + W_{friction} = \frac{1}{2} m v^{2}

with

W_{friction} < 0

. See Energy lost to friction.

"Tension in the pendulum string is the biggest force, so it must be feeding energy to the bob."

Error — tension points along the string, always perpendicular to the bob's velocity, so its work

F \cdot d = 0

; only gravity changes the bob's energy.

"At the lowest point of a swing the bob has maximum PE because it's moving fastest."

Error — fastest means maximum kinetic energy

K

; PE is at its minimum at the lowest point. The two trade off, they don't rise together.

"Spring PE is

U = \frac{1}{2} k x

Error — it's

U = \frac{1}{2} k x^{2}

; the force grows linearly with stretch

x

, so the stored energy (the integral of force) grows as

x^{2}

. See Spring potential energy (Hooke's law).

"Gravity does negative work as a ball falls, so PE increases."

Error — gravity does positive work on a falling ball (force and displacement both downward), so

W_{cons} > 0

and PE decreases. The sign of the work was flipped.

"Since energy is conserved, the block launched by a spring will keep that speed forever."

Error — conservation gives the speed at the launch instant; whether it stays constant depends on later forces (gravity, friction), which the single energy balance doesn't promise.

"I dropped a ball from

h

; it hits the ground at

v = g h

Error —

m g h = \frac{1}{2} m v^{2}

gives

v = 2 g h

; the factor of 2 was dropped.

Why questions

Why does the minus sign appear in

W_{cons} = - Δ U

So the force's work and the stored PE move in opposite directions: when the force does positive work (

W_{cons} > 0

), stored PE must drop (

Δ U < 0

Why is the energy method often easier than kinematics?

It compares only two states and ignores time and force direction, so you skip solving for the whole trajectory. See Work-energy theorem.

Why can't friction have a potential energy function?

Because friction's work

W_{friction}

depends on the path length (longer path = more heat), so it fails the path-independence test that defines a conservative force. See Conservative and non-conservative forces.

Why does the work–energy theorem hold even when forces are complicated?

It comes straight from Newton's 2nd law integrated over displacement; the algebra

F d x = m v d v

(see the glossary derivation above) never assumes the force is simple.

Why must we check that tension/normal forces do no work before invoking energy conservation?

Conservation of mechanical energy is only licensed when non-conservative or constraint forces contribute zero net work; confirming that is what makes the shortcut valid.

Why does a simple harmonic oscillator keep swapping KE and PE without losing total energy?

The spring force is conservative and (ideally) nothing else does work, so

K + U

is fixed while each rises and falls in turn. See Simple harmonic motion energy.

Why is power a different quantity from energy, not just a bigger version of it?

Power

P = \frac{d W}{d t}

is the rate of energy transfer; two processes can move the same energy but at very different powers. See Power.

Edge cases

An object is dropped from rest — what is its KE and total mechanical energy at the instant of release?

KE is zero (at rest) but total energy equals its PE

m g h

, which is nonzero; "at rest" does not mean "no energy".

A pendulum is released from the lowest point with zero speed — what happens?

Nothing moves — it is at the PE minimum with zero KE, a stable equilibrium; a small nudge just produces gentle restoring swings back toward that lowest point.

A spring is at its natural length and the block momentarily at rest — where is the energy?

Both spring PE and KE are zero at that instant; the energy is entirely in gravitational PE or was already dissipated — you must track where it went, not assume it vanished.

A ball is thrown up and comes back to the launch height — compare launch and return speeds (no air).

They are equal in magnitude, because

U

is the same at that height so

K

(hence speed) must match; only the direction of velocity has reversed.

At maximum spring compression, the block's speed is zero — is energy conserved at that instant?

Yes — KE is momentarily zero and all of it now lives as spring PE

\frac{1}{2} k x^{2}

; the total is unchanged, it just fully changed costume.

A mass sits still on a table — is its mechanical energy conserved?

Trivially yes; KE and (chosen) PE are both constant, and gravity's downward pull is balanced by the normal force doing zero work.

In deep space with no fields and no forces, does a drifting probe conserve mechanical energy?

Yes, but degenerately — PE is a constant (no field) and KE is constant (no force), so

K + U

is trivially frozen.

Recall One-line self-test

If someone hands you a problem and says "use energy conservation," what is the first thing you check? ::: Whether every non-conservative force (friction, air, applied pushes) does zero net work — if not, add its work as an explicit term instead of assuming $K + U$ is constant.

Question bank — Conservation of mechanical energy — derivation