Conservation of Mechanical Energy

A roller coaster speeds up as it descends because potential energy turns into kinetic energy. Without friction, total mechanical energy stays constant, so height alone can give you the speed. Think of the height energy at the top becoming motion energy lower down. Slide the height here and watch potential and kinetic energy trade places.

Energy Transformation

🎢 Secret of the Roller Coaster

①A roller coaster speeds up as it descends

②Height energy (potential) converts into speed energy (kinetic)!

③Without friction, total energy is constant → conserved

Visualizing Energy Conservation

Height h5 m

💡 Key Observations

①Top: 100% PE, 0% KE

②Bottom: 0% PE, 100% KE

③Everywhere PE + KE = constant! (energy conservation)

Deriving the Energy Formulas

Kinetic Energy

E_k = 12mv²

Energy of an object in motion

Gravitational Potential Energy

E_p = mgh

Energy of an object at height h

Conservation of Mechanical Energy

12mv₁² + mgh₁ = 12mv₂² + mgh₂

Without friction, total mechanical energy is the same anywhere

Applied to Free Fall

Speed at the Bottom After Falling from Height h

v = √(2gh)

mgh = ½mv² → v = √(2gh) (mass cancels!)

⚡ Mass Cancels Out!

①In energy conservation, mgh = ½mv² → m cancels from both sides

②Heavy or light, balls dropped from the same height reach the same speed!

③This is the heart of Galileo's Tower of Pisa experiment

Worked Examples

Example 1

An object is released from rest at a height of 20 m and falls freely. What is its speed on reaching the ground? (g = 10 m/s²)

From energy conservation mgh = ½mv², the mass cancels, giving v = √(2gh).

v = √(2gh)

Substitute g = 10, h = 20.

v = √(2 × 10 × 20) = √400 = 20 m/s

▸ 20 m/s

The landing speed v = √(2gh) is independent of mass. Heavy or light, the same height gives the same speed.

Example 2

An object is released from rest at a height of 5 m. At what height are its kinetic and potential energies equal?

Total energy equals the top potential energy mg(5). If KE = PE, each is half the total.

E_p = 12E_total = 12mg(5)

Solve mgh = ½mg·5 for h.

mgh = 12mg × 5 ⇒ h = 2.5 m

▸ 2.5 m

The point where KE = PE is always at half the drop height — independent of mass and g.

Summary

Kinetic Energy

E_k = 12mv²

Potential Energy

E_p = mgh

Energy Conservation (Core)

E_k + E_p = const (no friction)

Decrease in PE = Increase in KE

CSAT-style

On a frictionless incline, an object released from rest reaches a speed of 6 m/s at the bottom. What was its initial height? (g = 10 m/s²)

①1.2 m

②1.5 m

③1.8 m

④2.4 m

⑤3.6 m

▸ ③ 1.8 m

No friction, so mechanical energy is conserved: mgh = ½mv² → h = v²/(2g).

mgh = 12mv² ⇒ h = v²2g

Substitute v = 6, g = 10.

h = 6²2 × 10 = 3620 = 1.8 m

🎯 Exam Points

①Mechanical energy = KE + PE

②Conservation requires no non-conservative forces (friction, drag)

③Free-fall speed: v = √(2gh) — mass-independent!

④Pendulum: max speed at the bottom, speed 0 at the top

⑤With friction: E_k + E_p = E₀ - W_friction (some converts to heat)

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Momentum and Impulse

Heat and Thermodynamics

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