Energy Is Conserved, Even When It Doesn’t Look Like It
The law of conservation of energy says energy is never created or destroyed. It only changes form or moves from place to place. A rolling coaster car seems to break that rule: it starts with lots of energy and ends up stopped. But the energy didn’t vanish. Look at the whole ride in percent of the starting energy:
■ Potential■ Kinetic■ Thermal
Top of the lift hillPE 100% · KE 0% · thermal 0%
Bottom of the first dropPE 0% · KE 93% · thermal 7%
Car stopped at the stationPE 0% · KE 0% · thermal 100%
E_start = E_end + E_thermal
🔑When we say energy is “lost” to friction, we mean it’s no longer in a useful, organized form. It’s now thermal energy, spread among the atoms of the track, wheels, and air. The total is still 100%.
BMath
How Much Thermal Energy Does Friction Make?
In Activity 4-1-1 you saw that friction does negative work on a moving object: W = −F_f × d. That work doesn’t make energy disappear. The same amount shows up as thermal energy:
E_thermal = F_f × d
Put this into the energy budget and you get a single equation for any motion with friction:
PE_start + KE_start − F_f × d = PE_end + KE_end
Example 1: thermal energy on a ramp
Given
A 2.0 kg block starts at rest 0.50 m up a ramp that is 1.0 m long
It reaches the bottom with KE = 6.0 J
Solve
PE_start = m g h = 2.0 × 9.8 × 0.50 = 9.8 J
E_thermal = PE_start − KE_end = 9.8 − 6.0 = 3.8 J
F_f = E_thermal ÷ d = 3.8 ÷ 1.0
F_f = 3.8 N
✓ Friction made 3.8 J of thermal energy, with an average force of 3.8 N.
Example 2: speed at the bottom of a slide
Given
A 60 kg rider starts at rest 3.0 m up a slide
Friction produces 400 J of thermal energy on the way down
✓ The rider reaches the bottom at about 6.7 m/s. Without friction it would be about 7.7 m/s.
Example 3: where does a coasting coaster car’s energy go?
Given
A 500 kg car rolls at 8.0 m/s on a flat track
A friction force of 200 N acts until it stops
Solve
KE = ½ m v² = ½ × 500 × 8.0² = 16,000 J
All of it becomes thermal energy: F_f × d = KE
d = KE ÷ F_f = 16,000 ÷ 200
d = 80 m
✓ The car coasts 80 m, and 16,000 J of kinetic energy ends up as thermal energy.
CExample
Valley Decay Explorer
A marble in your valley rises a little less each crossing. Set the friction and see how the peaks shrink. Then see where the energy goes.
Each bar is how high the marble gets after another crossing of the valley. Raise the friction and watch it run out of climb faster. Then slide the crossing count and watch the thermal share grow.
Peak after 3 crossings: 18.4 cm
Still PE: 61%Now thermal: 39%
The total is always 100%. The marble’s motion fades, and the energy goes into the thermal slice.
DMath
Budgeting a Climb
Here is the method you’ll use in Project 4-2-3. Measure the fraction f of its release height that the marble returns. Then, to reach a hill of height H, release it from a height that leaves exactly H after friction takes its share:
f = h₂ ÷ h₁ h₁ = H ÷ f
A hill budget
Given
Marble mass 0.005 kg
In the valley it was released from 0.20 m and rose to 0.17 m
Target hill: H = 0.15 m
Solve
f = 0.17 ÷ 0.20 = 0.85
h₁ = H ÷ f = 0.15 ÷ 0.85 = 0.176 m
Energy at release = 0.005 × 9.8 × 0.176 = 8.65 mJ
Energy needed at the hill = 0.005 × 9.8 × 0.15 = 7.35 mJ
Friction takes 8.65 − 7.35 = 1.30 mJ (15% of the start)
✓ Release from about 0.176 m. Friction takes 1.3 mJ, and the other 7.35 mJ gets the marble to the hill.
⚠️This method assumes friction takes the same fraction of the energy at every release height. That’s a good approximation for a smooth valley, but not perfect. A bump, a tight curve, or a very different slope will change f, so test your prediction. If the marble must cross more than one valley, the loss compounds: the second crossing returns f of what’s left, so after two crossings the height is f² of the original.
EWatch Out
Three Common Mix-Ups
⚠️“Friction destroys energy.” It doesn’t. Friction converts mechanical energy into thermal energy. The total energy is exactly the same. It’s just no longer in a form that can move the marble or the car.
⚠️“Mechanical energy (PE + KE) is always conserved.” Only when no friction or similar force acts. With friction, you have to add the thermal energy to get a total that stays constant.
⚠️“Friction is always bad.” You need it to walk, to stop a car, and to keep a coaster on the track. Engineers aim to control friction. They seldom want zero.
FConcept
Efficiency
Not all of the energy you put into a machine ends up doing the job. The efficiency of a machine compares the useful energy it delivers to the total energy it takes in:
η = E_out ÷ E_in × 100%
The leftover, E_in − E_out, is the wasted energy, almost always thermal energy. Here are some rough typical values:
Machine
Typical efficiency
Large electric motor
about 90%
Hydroelectric turbine
about 90%
Gasoline car engine
about 25–30%
Human muscle
about 25%
Your marble valley (height returned)
about 70–95%
💡Treat these as ballpark figures. Real efficiencies depend on the design, size, and how the machine is used.
GExample
Efficiency Meter
Try a few efficiencies and watch the wasted slice. Then try to find a way to make it disappear.
Set how much energy goes in and how efficient the machine is. The amber slice is energy that became thermal energy instead of useful output.
η = 350 ÷ 500 × 100% = 70%
Useful out: 350 JWasted as heat: 150 J
Even a very good machine wastes some energy as heat.
HMath
Efficiency Calculations
Example 1: a lifting motor
Given
A motor takes in 500 J of electrical energy
It lifts a load, which gains 350 J of PE
Solve
η = E_out ÷ E_in × 100% = 350 ÷ 500 × 100%
η = 70%
Wasted = 500 − 350 = 150 J
✓ The motor is 70% efficient and wastes 150 J as heat.
Example 2: how much input is needed?
Given
A hoist is 80% efficient
It lifts a 50 kg load 10 m
Solve
E_out = m g h = 50 × 9.8 × 10 = 4,900 J
E_in = E_out ÷ η = 4,900 ÷ 0.80
E_in = 6,125 J
✓ The hoist has to take in 6,125 J to deliver 4,900 J of useful lift.
Example 3: a car engine
Given
An engine burns fuel that releases 100,000 J
25,000 J of that moves the car
Solve
η = 25,000 ÷ 100,000 × 100% = 25%
Wasted = 100,000 − 25,000 = 75,000 J
✓ Three-quarters of the fuel’s energy ends up as heat in the engine, exhaust, and surroundings.
IConcept
Why No Machine Is 100% Efficient
Any machine with moving parts has friction somewhere: in bearings, against air, between gears. Every bit of friction turns some organized energy into thermal energy. You can reduce it with smoother parts and lubricants, but you can’t remove it completely.
There’s a deeper reason too. Thermal energy is the random jiggling of vast numbers of particles. Once energy is spread out that way, gathering all of it back into one neat, organized motion, like a marble rolling in one direction, does not happen on its own. Heat engines can turn some thermal energy back into work, but never all of it. That is why the “missing” energy is permanent in practice, and why even the best machine falls short of 100%.
🔑Energy is conserved, but useful energy isn’t. Every real process leaves some energy in a form that’s too spread out to use. Engineers design to waste as little as possible, and they budget for what’s left.
JConcept
Quick Check
Four questions. They are not graded. Answer them to find out what still needs work.
Q1A coaster car rolls to a stop on a flat track. Where did its kinetic energy go?
Q2A 2.0 kg block is released from rest 0.50 m up a ramp and has 6.0 J of KE at the bottom. How much thermal energy did friction produce?
Q3A machine takes in 800 J of energy and delivers 600 J of useful work. What is its efficiency?
Q4Why can’t any real machine be 100% efficient?
Score: 0 of 4
KConcept
Looking Ahead
You now have the full toolkit for the unit: work, KE, PE, power, and an energy budget that includes friction. Next you’ll put it to use.