Law of Conservation of Energy: Statement, Derivation and Examples

The law of conservation of energy states that energy can neither be created nor destroyed; it can only be changed from one form to another, so the total energy of an isolated system stays constant. In mechanics, when only conservative forces such as gravity or a spring force do work, this becomes the principle of conservation of mechanical energy: kinetic energy + potential energy = constant. The rest of this page gives the statement in exam form, the standard derivation for a freely falling body, worked numbers and the general thermodynamic form.

Law of conservation of energy statement

Two versions are used in NCERT-based exams, and it helps to know which one a question wants:

• General statement (Class 9 style): Energy can neither be created nor destroyed. It can only be transformed from one form to another, and the total energy before and after the transformation remains the same.
• Principle of conservation of mechanical energy (Class 11 style): The total mechanical energy of a system is conserved if the forces doing work on it are conservative.

If a question says “state the principle of conservation of energy”, write the general statement and add one example, such as a falling ball turning potential energy into kinetic energy.

Mathematical form

For an isolated system: E_total = constant, or ΔE_total = 0.

For mechanical energy under conservative forces:

K_i + U_i = K_f + U_f, where K = ½mv² and, near the Earth’s surface, U = mgh.

Derivation of the law of conservation of energy for a freely falling body

This is the derivation most often asked in board exams. Take a body of mass m held at rest at point A, at height h above the ground. It falls freely (air resistance ignored). We check the total energy at three points.

At point A (top, height h)

• Velocity u = 0, so KE = ½m(0)² = 0
• PE = mgh
• Total energy E_A = 0 + mgh = mgh

At point B (after falling a distance x)

• Height above ground = h – x, so PE = mg(h – x)
• From v² = u² + 2as with u = 0, a = g and s = x: v² = 2gx
• KE = ½m(2gx) = mgx
• Total energy E_B = mg(h – x) + mgx = mgh

At point C (just before touching the ground)

• Height = 0, so PE = 0
• Distance fallen = h, so v² = 2gh
• KE = ½m(2gh) = mgh
• Total energy E_C = mgh + 0 = mgh

Since E_A = E_B = E_C = mgh, the total mechanical energy stays the same at every point of the fall. Potential energy lost is exactly equal to kinetic energy gained. That proves the law for a freely falling body.

Solved example: 2 kg ball dropped from 20 m

Take m = 2 kg, h = 20 m, g = 9.8 m/s², and point B after falling x = 5 m.

Check at C: ½ x 2 x 19.8² = 392 J. The total is 392 J throughout.

In real air the ball arrives a little slower. If it were measured at 18 m/s, its KE would be ½ x 2 x 18² = 324 J, so 392 – 324 = 68 J went into heating and moving the air. The energy was not destroyed; it left the mechanical account.

Example: a simple pendulum

Pull a pendulum bob aside and it rises a little. At the end of the swing it stops for an instant: all potential energy, no kinetic energy. At the lowest point it moves fastest: all kinetic energy (taking that point as zero height).

Say a 0.5 kg bob is released 0.2 m above its lowest point. Its PE there is 0.5 x 9.8 x 0.2 = 0.98 J. At the bottom, ½mv² = 0.98 J, so v = √(2gh) = √(2 x 9.8 x 0.2) = 1.98 m/s. Notice the mass cancels, so any bob released from 0.2 m reaches the same speed. A real pendulum slowly dies down because air drag and friction at the pivot turn its energy into heat. Newton’s cradle, shown above, shows the same exchange passing from ball to ball.

Example: a roller coaster

A roller coaster has no engine after the first climb. A chain lifts the car to the highest point, and from then on it runs on the potential energy stored there.

A 500 kg car starts almost from rest at the top of a 45 m hill. Stored energy = 500 x 9.8 x 45 = 220,500 J. Ignoring friction:

• At ground level: v = √(2 x 9.8 x 45) = 29.7 m/s (about 107 km/h)
• At the top of a second hill 25 m high (a drop of 20 m): v = √(2 x 9.8 x 20) = 19.8 m/s

This is why every later hill must be lower than the first one: the car can never climb higher than its starting height, and friction takes a share on every stretch of track.

Energy transformations in daily life and engineering

In every row, energy in = useful energy out + wasted energy. Efficiency is only the fraction that ends up in the form you want. For a hydro plant, 1 m³ of water (1,000 kg) falling 100 m carries 1,000 x 9.8 x 100 = 980 kJ; a plant converting 90% of this would deliver 882 kJ as electricity, and the other 98 kJ would still exist as heat. Read more about where geothermal energy comes from. The same bookkeeping applied to charge in a circuit gives Kirchhoff’s voltage law, which is conservation of energy around a loop.

Link with the work-energy theorem

The work-energy theorem says the net work done on a body equals its change in kinetic energy: W_net = ΔK. Split that work into the part done by conservative forces (W_c = -ΔU) and the part done by non-conservative forces such as friction (W_nc):

ΔK + ΔU = W_nc

When W_nc = 0, Δ(K + U) = 0 and mechanical energy is conserved. When friction acts, W_nc is negative and mechanical energy falls by exactly the amount turned into heat, as in the 68 J example above. For more on the forms involved, see our page on mechanical energy.

General form: the first law of thermodynamics

Once heat is included, conservation of energy becomes the first law of thermodynamics:

ΔU = Q – W

• ΔU = change in internal energy of the system
• Q = heat supplied to the system (positive when heat flows in)
• W = work done by the system (positive when the gas expands and pushes on its surroundings)

Example: a gas absorbs 500 J of heat and does 200 J of work pushing a piston. ΔU = 500 – 200 = 300 J. This sign convention is the one used in physics and mechanical engineering. Chemistry books usually write ΔU = q + w, with w as work done on the system; both say the same thing. Heat engines such as the Otto cycle and the ideal Carnot cycle are analysed with exactly this energy balance.

Mass-energy: E = mc²

Einstein’s special relativity (1905) showed that mass itself is a form of energy: E = mc², with c = 3 x 10⁸ m/s. Converting just 1 g of mass gives 0.001 x (3 x 10⁸)² = 9 x 10¹³ J, about 25 million kWh. In nuclear reactions the products weigh slightly less than the reactants, and that “missing” mass appears as energy. So the modern statement is conservation of mass-energy; the law was never broken, it just needed mass counted as energy.

Why perpetual motion machines are impossible

A perpetual motion machine of the first kind would give out more energy than it takes in, or keep running forever with no input. That directly violates the law of conservation of energy. A machine of the second kind would turn heat from a single source completely into work with no other effect; it breaks no energy balance but violates the second law of thermodynamics. Every real machine loses some energy to friction as heat, so without a supply it always slows down and stops.

Common exam mistakes

• Saying “energy is lost” to friction. It is converted to heat, not destroyed. Write “converted to heat” in answers.
• Mixing up the two statements. Total energy is always conserved; mechanical energy is conserved only when no friction or other non-conservative force does work.
• Forgetting the reference level for PE. Choose zero height once and use it for every point.
• Using v = u + at when the path is curved (pendulum, coaster). Energy methods need only the heights, not the path.
• Wrong sign for W in the first law. State your convention in the answer.

History of the law

The idea grew out of work on heat in the 1840s. The German physician Julius Robert von Mayer stated the equivalence of heat and mechanical work in 1842. In 1843 James Prescott Joule independently measured the mechanical equivalent of heat in a series of experiments. In 1847 Hermann von Helmholtz set out the principle in general mathematical form in his essay Uber die Erhaltung der Kraft (On the Conservation of Force). For the school treatment, see NCERT’s chapter on work, energy and power at ncert.nic.in.

FAQs

What is the law of conservation of energy statement?

Energy can neither be created nor destroyed; it can only be converted from one form to another. The total energy of an isolated system remains constant.

How do you derive the law of conservation of energy?

Take a body of mass m falling freely from height h. At the top, total energy = 0 + mgh. After falling x, it is mg(h – x) + mgx = mgh. At the ground, it is mgh + 0. The total is mgh at every point, so mechanical energy is conserved.

Is mechanical energy always conserved?

No. Mechanical energy is conserved only when conservative forces such as gravity or a spring force do the work. Friction and air drag convert part of it to heat, although total energy is still conserved.

What is an example of conservation of energy?

A 2 kg ball dropped from 20 m has 392 J of potential energy at the top and 392 J of kinetic energy just before landing, when it moves at 19.8 m/s. A swinging pendulum and a hydroelectric plant are other examples.

Who discovered the law of conservation of energy?

It was developed in the 1840s by Julius Robert von Mayer (1842), James Prescott Joule (1843) and Hermann von Helmholtz (1847), who gave it a general mathematical form.

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