Once a roller coaster has been hauled up to its highest point, it climbs and dives over hill after hill without an engine. The potential energy stored at the first height turns into kinetic energy on the way down and back into potential energy on the way up.
The sum of potential energy (mgh) and kinetic energy (½mv²) is mechanical energy, and with no friction or air resistance it is the same at every point.
Moving the train in the first diagram, the kinetic energy bar grows the lower it goes, while the two bars together stay the same length; that is why it can never get over a hill higher than where it started.
The second diagram shows how the two kinds of energy are shared at each height for an object released from rest at 50 m.
Where Does the Energy Go?
🎢 Seen on a Roller Coaster
①A roller coaster speeds up as it runs down from a high point to a low one
②The potential energy it had from being high has turned into kinetic energy of motion
③Climbing the next hill, kinetic energy turns back into potential energy and it slows down
④Energy is not lost; it only changes form
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Potential and kinetic energy trading places as a roller coaster moves along the track
10%
Potential and Kinetic Energy
⚡ Two Kinds of Mechanical Energy
①Potential energy grows with height: Eₚ = mgh (weight × height)
②Kinetic energy grows with speed: Eₖ = ½mv²
③Going lower, potential energy falls and kinetic energy rises by the same amount, so it speeds up
④Going higher, potential energy rises and kinetic energy falls by the same amount, so it slows down
⑤The graph shows the share of each energy at each height for an object released from rest at 50 m
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Bar graph of potential, kinetic and total mechanical energy at different heights
40m
Conservation of Mechanical Energy
⚖️ The Total Stays the Same
①When there is no friction or air resistance
②potential energy + kinetic energy = constant
③This is the law of conservation of mechanical energy
④At the highest point, potential energy is greatest; if the object is at rest there, kinetic energy is 0
⑤At the lowest point, potential energy is smallest and kinetic energy greatest
Summary
Conservation of mechanical energy
Potential energy + kinetic energy = constant
Without friction, potential plus kinetic energy is the same everywhere
Potential Energy
PE = mgh
mass × gravity × height
Kinetic energy
KE = 12mv²
½ × mass (m) × speed² (v²)
🎯 Common Test Points
①Ignoring friction, potential energy + kinetic energy = constant
②High up: large potential energy, small kinetic energy
③Low down: small potential energy, large kinetic energy
④While falling, the potential energy lost equals the kinetic energy gained
⑤In reality friction slowly reduces mechanical energy, turning the lost amount into heat
Examples and Unit Review
Example 1
A roller coaster starts from rest at its highest point with no friction. What are its potential and kinetic energy at the highest and the lowest points?
1
At the highest point the height is greatest, so potential energy is greatest, and since it is at rest, kinetic energy is 0.
2
At the lowest point potential energy is smallest, and kinetic energy has grown by the amount of potential energy lost, so it is fastest.
▸ Highest point: maximum potential energy, kinetic energy 0. Lowest point: minimum potential energy, maximum kinetic energy
Starting from 50 m, as it passes 20 m it keeps 40 % of its original potential energy and 60 % has become kinetic energy.
Example 2
With no friction or air resistance, what happens to the sum of potential and kinetic energy while an object falls?
1
Without friction or air resistance, kinetic energy increases by exactly as much as potential energy decreases.
2
So the mechanical energy, the sum of the two, is the same at every height.
▸ The sum does not change (conservation of mechanical energy)
Real roller coasters lose a little energy to friction, which is why each later hill is built lower.
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
1
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).
2
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).