Grade 10-11 (age 15-17)

high school physics Conservation of Mechanical Energy

A roller coaster speeds up as it comes down from a height because potential energy is turning into kinetic energy. Without friction or air resistance their sum, the mechanical energy, stays exactly as it started, so the height alone gives the speed at the bottom. The potential energy held at the top passes bit by bit into kinetic energy on the way down, and at the bottom all of it is kinetic. Move the height slider and the bars for the two energies trade places at the top, middle and bottom, while their sum always stays the same.

Energy Changing Form

🎢 The secret of the roller coaster

①A roller coaster goes faster the further it comes down

②Potential energy from height turns into kinetic energy from speed

③Without friction, the sum of the two energies is always the same

④The figure uses mass m = 1 kg and g = 9.8 m/s²

⑤With friction, part of the mechanical energy becomes heat

Conservation of Energy

Height h5 m

💡 What to notice

①At the top all the energy is potential and the kinetic energy is 0

②In the middle, at half the height, the two energies are half each

③At the bottom the potential energy is 0 and all of it is kinetic

④Everywhere, potential energy + kinetic energy is the same

Energy Formulas

Kinetic energy

E_k = 12mv²

The energy of a moving object

Gravitational potential energy

E_p = mgh

The energy of an object at height h above the reference level

Conservation of mechanical energy

12mv₁² + mgh₁ = 12mv₂² + mgh₂

Without friction, the mechanical energy is the same at every position

Applying It to Free Fall

Speed at the bottom after falling from height h

v = √(2gh)

In mgh = ½mv² the mass m cancels from both sides, giving v = √(2gh)

⚡ The mass cancels

①Both sides of mgh = ½mv² contain m, so it cancels

②So, ignoring air resistance, a heavy ball and a light ball dropped from the same height land at the same speed

③This is the idea behind Galileo's famous Leaning Tower of Pisa story

④A pendulum works the same way: fastest at the lowest point and momentarily at rest at the highest

Worked Examples

Example 1

An object is released from rest at a height of 20 m and falls freely. How fast is it moving as it reaches the ground? Use g = 10 m/s².

In mgh = ½mv² the mass cancels, so v = √(2gh).

v = √(2gh)

Put in g = 10 and h = 20.

v = √(2 × 10 × 20) = √400 = 20 m/s

▸ 20 m/s

The speed at the bottom, √(2gh), does not depend on the mass.

Example 2

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

The starting energy is the potential energy at the top, mg × 5. When kinetic and potential energy are equal, each is half of that.

E_p = 12E_total = 12mg(5)

Solve mgh = ½ × mg × 5 for h.

mgh = 12mg × 5 ⇒ h = 2.5 m

▸ 2.5 m

The two energies are equal at half the starting height every time, whatever the mass and g.

Summary

Kinetic energy

E_k = 12mv²

Potential energy

E_p = mgh

Conservation of energy

E_k + E_p = const (no friction)

The kinetic energy gains exactly what the potential energy loses

CSAT-style problem, adapted

An object is released from rest on a frictionless slope and reaches 6 m/s at the bottom. What was its starting height? Use g = 10 m/s².

①1.2 m

②1.5 m

③1.8 m

④2.4 m

⑤3.6 m

▸ ③ 1.8 m

There is no friction, so mgh = ½mv², giving h = v²/(2g).

mgh = 12mv² ⇒ h = v²2g

Put in v = 6 and g = 10.

h = 6²2 × 10 = 3620 = 1.8 m

🎯 Exam Points

①Mechanical energy = kinetic energy + potential energy

②It is conserved when there is no friction or air resistance

③The speed after falling from height h is v = √(2gh), whatever the mass

④A pendulum is fastest at its lowest point and stops at its highest

⑤With friction, the mechanical energy drops by the work done by friction, which becomes heat

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Momentum and Impulse

Heat and Laws of Thermodynamics

Last updated 2026-10-17

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