Conservation of Mechanical Energy

Mechanical energy is the sum of potential and kinetic energy, and without friction that sum stays constant. When a roller coaster comes down from a high point, potential energy falls and kinetic energy rises by the same amount, so speed increases. Potential energy is PE = mgh and kinetic energy is KE = ½mv²; at the top PE is max and KE is 0, and at the bottom it is the reverse. On this page, change position and height to check that the two energies trade places while their sum stays the same.

Where Does the Energy Go?

🎢 Intuition — Roller Coaster

①A coaster starting from the top speeds up at the bottom

②Height energy (PE) becomes speed energy (KE)

③Energy is not lost — only its form changes!

Position10%

Potential vs Kinetic Energy

⚡ Two Mechanical Energies

①PE: bigger when higher → PE = mgh

②KE: bigger when faster → KE = ½mv²

③Lower height → PE↓ → KE↑ (faster)

④Higher height → PE↑ → KE↓ (slower)

Height h40m

Conservation Law

⚖️ The Sum is Constant

①Without friction / air resistance

②PE + KE = constant

③That is the conservation of mechanical energy

④Top: PE max, KE = 0

⑤Bottom: PE = 0, KE max

Summary

Conservation of Mechanical Energy

PE + KE = const → mgh + 12mv² = const

PE + KE = always the same value

Potential Energy

PE = mgh

mass × gravity × height

Kinetic Energy

KE = 12mv²

½ × mass × speed²

🎯 Exam Points

①PE + KE = constant (ignoring friction)

②Higher → bigger PE, smaller KE

③Lower → smaller PE, bigger KE

④At top with v=0, KE = 0

⑤In reality friction converts some to heat

Examples and Unit Test

Example 1

A roller coaster starts, without friction, at the highest point. At the highest and lowest points, what happens to the potential energy (PE) and the kinetic energy (KE)?

At the highest point the height is greatest so PE is maximum, and if it starts with zero speed then KE = 0.

At the lowest point the height is zero so PE = 0, and KE becomes maximum by the same amount.

▸ Highest point: PE maximum, KE = 0 / lowest point: PE = 0, KE maximum

As the height drops, PE turns into KE and the speed increases.

Example 2

With no friction or air resistance, how does the sum of potential and kinetic energy (the mechanical energy) change as an object falls?

Without friction or air resistance, KE increases by exactly the amount PE decreases.

So PE + KE, the mechanical energy, is always conserved at the same value at every position.

▸ PE + KE = constant (conservation of mechanical energy)

Without friction the energy only changes form while the total is conserved.

Unit Test

Which statement about the conservation of mechanical energy is correct?

①The lower an object goes, the greater its potential energy

②Without friction, the sum of potential and kinetic energy is constant

③Kinetic energy is greatest at the highest point

④When kinetic energy increases, the mechanical energy increases with it

⑤Even with real friction, mechanical energy is conserved

▸ ② Without friction, the sum of potential and kinetic energy is constant

The higher the position, the greater the potential energy (① wrong), and at the highest point KE is zero while PE is greatest (③ wrong).

Without friction KE rises by as much as PE falls, so the sum is constant (④ wrong, the sum is conserved), and with real friction mechanical energy turns into heat and decreases (⑤ wrong); without friction PE + KE is constant (② correct).

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Thermal Energy

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