Engineering Exams6 min readAug 30, 2026

Conservation of Energy: The Universe’s Perfect Bookkeeping

6 min read · 1,024 words

JEE/NEET Physics · Work, Energy & Power series · Part 4 of 8 · All parts →

The One Rule
The Famous Result: v = √(2gh), No Mass Anywhere
When Friction Leaks the Ledger
The Pendulum’s Endless Trade
Solved Examples
This Physics in Your Daily Life
Practice set (answers hidden — try first)

✪ Key points — the 30-second version

Total energy (KE + PE) stays constant when only conservative forces act
Falling: mgh converts exactly to ½mv² — v = √(2gh), mass cancels!
With friction: ME_lost = friction force × distance (the leak is measurable)
Pendulum and rollercoaster: endless PE↔KE trading
Energy is never destroyed — only moved or downgraded

Drop anything — a feather (in vacuum) or an elephant — from the same height, and both hit the ground at the same speed. Mass doesn’t even enter the answer. That’s energy conservation at work: the universe’s most reliable bookkeeping. Part 4 of the Work, Energy & Power series.

In this card

The one rule
What each letter means
The famous result: v = √(2gh), no mass anywhere
When friction leaks the ledger
The pendulum’s endless trade
Solved examples
Common mistakes
This physics in your daily life
Practice set
Recap

The One Rule

The energy see-saw: what motion loses, height gains — the total never changes (until friction leaks it as heat)

KE + PE = constant (when only gravity/springs act)motion energy + stored energy = unchanging total

Letter	What it means (plain words)	Value / unit
KE	motion energy ½mv²	J
PE	stored energy: mgh (height) and/or ½kx² (spring)	J
friction (if present)	the leak: total drops by friction × distance	the only common spoiler

Read it as a see-saw: what KE loses, PE gains, exactly. Total never changes (with only gravity/springs). With friction, the total still doesn’t vanish — it leaks out as heat: mechanical energy lost = friction force × distance slid.

The Famous Result: v = √(2gh), No Mass Anywhere

Drop from height h: mgh = ½mv² → divide both sides by m — mass cancels completely → v = √(2gh). Heavy or light, same landing speed (in vacuum). From 20 m: v = √400 = 20 m/s. From 45 m (with g = 10): 30 m/s. One line, no mass, no time — the most useful result in the chapter.

When Friction Leaks the Ledger

Real slides and roads have friction. The bookkeeping then reads: (KE + PE)_start = (KE + PE)_end + friction × distance. The leak isn’t lost — it’s heat (why brake discs glow, why rubbing warms hands). Questions love this: ‘how far does it slide before stopping?’ — the leak formula answers in one line.

The Pendulum’s Endless Trade

A pendulum swings because energy endlessly converts: maximum height (all PE, still) → bottom (all KE, fastest) → the other side’s height (all PE again). With zero friction it would swing forever; real pendulums leak tiny heat each swing — that’s why clocks needed winding.

Solved Examples

✎ Easy — the drop. Speed after falling 45 m (no air, g = 10)?

Famous result: v = √(2gh) = √(2 × 10 × 45) = √900.

v = 30 m/s — no mass needed, ever. ✔

Answer: 30 m/s

✎ Exam level — the ramp with friction. A 2 kg block slides from rest down a 3 m ramp (angle: height = 1.5 m) with friction 4 N acting along a 3 m path. Speed at the bottom (g = 10)?

Ledger: start PE = 2 × 10 × 1.5 = 30 J. Leak = friction × distance = 4 × 3 = 12 J. Remaining for KE = 18 J.

½(2)v² = 18 → v = √18 ≈ 4.24 m/s.

Check: without friction it’d be √30 ≈ 5.48 — friction slowed it, as it must. ✔

Answer: v ≈ 4.24 m/s

✎ JEE level — the loop. A bead slides from rest at height h on a frictionless track with a vertical loop of radius R at the bottom. Minimum h to complete the loop?

Two conditions meet: at the loop’s top, gravity supplies the needed centripetal push: mg = mv²/R → v²_top = gR. Energy: mg·h = mg·(2R) + ½m·gR → h = 2R + R/2.

h = 2.5R.

This is the classic rollercoaster design number — five-halves the loop radius (in practice more, for friction). ✔

Answer: h = 2.5R (the rollercoaster rule)

⚠ Mistakes students make — and how to avoid them

Putting mass in the drop formula. v = √(2gh) has no mass — inserting one means the algebra was never finished.
Forgetting the friction leak term. ‘Energy is conserved’ is FALSE with friction present; mechanical energy falls by friction × distance.
Height measured inconsistently. Keep one zero level for the whole problem (Part 3’s rule).
Believing energy conservation means nothing is lost ever. Energy is never destroyed — but it DOWNGRADES to heat, which is usually unusable. The ledger always balances; usefulness doesn’t.

This Physics in Your Daily Life

◎ This physics in your daily life

Every rollercoaster’s first hill is its battery — the rest of the ride spends that mgh. Engineers add margin above 2.5R for friction.
Hydro dams again, quantitatively: 1,000 tonnes falling 100 m delivers ~1 billion joules — v = √(2gh) for the water, then turbines take over.
Regenerative braking: EVs intercept the KE you’d normally burn in brakes and bank it into the battery — conservation, monetised.
A swing in the park: you pump by leaning at the right moments (adding small energy each cycle); friction and air take tiny tolls — the trade is visible physics.
Meteors burn up because v is enormous: ½mv² at 30 km/s converts to heat on air contact — conservation you can watch as a shooting star.

Practice set (answers hidden — try first)

(NEET-level) Speed after a 20 m free fall (g = 10):

√(2×10×20) = 20 m/s.

(JEE Main-level) A 1 kg block slides 5 m on flat ground against friction 6 N, starting at 8 m/s. It stops after:

KE = ½(1)(64) = 32 J = 6 × d → d ≈ 5.33 m.

(Concept) A pendulum’s speed is maximum at:

The lowest point — all PE traded into KE.

(JEE Main-level) Loop-the-loop minimum release height (loop radius R, frictionless):

2.5R.

(Concept) With friction present, ‘energy is conserved’ — true or false?

Mechanical energy: false (leaks as heat). Total energy including heat: always true.

🧠 Memory tricks & everyday anchors — the 20-second revision

🧠 Chant: ‘what motion loses, height gains — the total never changes’.
🧠 The mass-free line: ‘drop questions don’t need mass — √2gh and done’.
🧠 2.5R: the rollercoaster rule — say it like a phone number.
🏠 Daily: every coaster’s first hill is the ride’s battery; everything after is spending.
🏠 Daily: a shooting star is ½mv² turned to heat in front of your eyes.
🔁 KE + PE constant under gravity/springs
🔁 v = √(2gh): no mass, no time needed
🔁 friction leak = friction × distance → heat

▶ Recap card — save for revision week