seegongsik
Saved words
Grade 10-11 (age 15-17)

high school physics Conservation of Mechanical Energy

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. Change the height and 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
④This sketch uses m = 1 kg
⑤with friction, some mechanical energy becomes heat

Visualizing Energy Conservation

5 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
Ek = 12mv²
Energy of an object in motion
Gravitational Potential Energy
Ep = 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
④A pendulum is the same: max speed at the bottom, speed 0 at the top

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)
1
From energy conservation mgh = ½mv², the mass cancels, giving v = √(2gh).
v = √(2gh)
2
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?
1
Total energy equals the top potential energy mg(5). If KE = PE, each is half the total.
Ep = 12Etotal = 12mg(5)
2
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
Ek = 12mv²
Potential Energy
Ep = mgh
Energy Conservation (Core)
Ek + Ep = const (no friction)
Decrease in PE = Increase in KE
exam-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
1
No friction, so mechanical energy is conserved: mgh = ½mv² → h = v²/(2g).
mgh = 12mv2 ⇒ h = v22g
2
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)
← Previous
Momentum and Impulse
Next →
Heat and Laws of Thermodynamics
Was this helpful? Support seegongsik