Understand mass-energy equivalence the easy way, with plain English intuition, a clear diagram, the rest energy and kinetic energy equations, worked examples and an auto marked practice test. VCE Physics Units 3 and 4.
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Albert Einstein discovered something that sounds impossible at first. Mass and energy are the same thing, just wearing different clothes. A lump of matter sitting perfectly still is secretly a huge store of energy, and the recipe that converts one into the other is the most famous equation in all of science: E0=mc2.
Mass is frozen energy
Think of mass as energy that has been frozen solid. Even when an object is not moving at all, it still holds a vast amount of energy locked inside it. We call this its rest energy, and it depends only on the mass:
E0=mc2
The reason the number is so enormous is the c2 part. The speed of light c is 3.0×108 m/s, and squaring it gives 9.0×1016. So you multiply the mass by ninety thousand million million. A tiny mass becomes a colossal energy.
Moving objects carry even more
Once an object starts moving, it carries its rest energy plus extra energy from its motion. The total energy is the rest energy stretched by the Lorentz factorγ, a number that grows as the object speeds up:
Etot=γmc2
The leftover, the part that is purely due to movement, is the kinetic energy:
Ek=(γ−1)mc2=Etot−E0
At everyday speeds γ is almost exactly 1, so Ek is tiny and matches the school formula. Near the speed of light γ shoots up, and the kinetic energy becomes huge.
When mass becomes pure energy
The cleanest demonstration of mass-energy equivalence is annihilation. When an electron meets its antimatter twin, a positron, the two particles vanish completely. All of their mass is converted into energy, carried away as two photons of light flying off in opposite directions.
The blue circle is the electron and the red circle is the positron. They drift together, touch, and disappear. In their place, two photons shoot off in opposite directions, carrying away every bit of energy that used to be mass. Nothing is destroyed, the mass is simply turned into energy.
How to actually solve one
The method is a short, repeatable recipe.
For rest energy, square the speed of light first (c2=9.0×1016), then multiply by the mass: E0=mc2.
For total energy of a moving object, multiply the rest energy by the Lorentz factor: Etot=γE0.
For kinetic energy, take the rest energy back out: Ek=Etot−E0=(γ−1)E0.
Keep your powers of ten lined up when you add or subtract energies.
Watch the trap. Total energy includes the rest energy, but kinetic energy does not. If a question asks only for the energy of motion, remember to subtract E0.
Lock it in with active recall
Cover the answer and say each one out loud before you flip. Rate yourself honestly — the cards you find hard come back sooner, the ones you know are spaced further out.
Active recall
Answer from memory first, then flip. Rate yourself and each card returns on a spaced schedule (1 → 3 → 7 → 16 days).
Write the rest energy of an object of mass m.
E0=mc2. The speed of light is squared, which is why even a tiny mass holds enormous energy.
Why is the energy from a tiny mass so large?
Because the mass is multiplied by c2=9.0×1016 — about ninety thousand million million — so a tiny mass becomes a colossal energy.
Write the total energy of a moving object.
Etot=γmc2=γE0 — the rest energy stretched by the Lorentz factor.
Write the kinetic energy of a fast particle two ways.
Ek=(γ−1)mc2=Etot−E0 — the total energy with the rest energy taken back out.
What is the difference between total and kinetic energy?
Total energy is everything (rest + kinetic); kinetic energy is only the extra energy due to motion. Don’t forget to subtract E0 for kinetic.
What happens in electron–positron annihilation?
The two particles vanish and all their mass converts into energy, carried off as two photons flying in opposite directions.
Recall · Special Relativity
Write the Lorentz factorγ.
γ=1−v2/c21, which appears in both time dilation and the total energy Etot=γmc2.
Recall · Atomic Spectra and Energy Levels
Where does the energy of an emitted photon come from in an atom?
From the gap between two energy levels: ΔE=hf=Ehigh−Elow — energy released as the electron drops down.
See the recipe in action in the Worked Examples tab, then test yourself in Try It.
Worked examples
Worked Example 1The rest energy of an electron
Find the rest energy of an electron, which has a mass of 9.1×10−31 kg. Take c=3.0×108 m/s.
1
Rest energy is just the mass multiplied by the speed of light squared. Start by squaring c.
c2=(3.0×108)2=9.0×1016 m2/s2
2
Now multiply the mass by that number.
E0=mc2=9.1×10−31×9.0×1016
3
Multiply the front numbers, then add the powers of ten.
E0=8.2×10−14 J
Answer
E0=8.2×10−14 J
Worked Example 2Total and kinetic energy of a fast particle
A particle has a rest energy of 8.0×10−14 J and moves at a speed where the Lorentz factor is γ=2.0. Find its total energy and its kinetic energy.
1
Total energy is the rest energy stretched by the Lorentz factor. Just multiply the two together.
Etot=γE0=2.0×8.0×10−14=1.6×10−13 J
2
Kinetic energy is whatever total energy is left over once you take the rest energy back out.
Ek=Etot−E0=1.6×10−13−8.0×10−14
3
Line up the powers of ten and subtract.
Ek=1.6×10−13−0.80×10−13=8.0×10−14 J
Answer
Etot=1.6×10−13 J,Ek=8.0×10−14 J
Practice questions
Practice test
Try it yourself
5 questions, 7 marks
Choose your answers, then submit to see your score and the full worked solutions.
Multiple choice is marked for you, just like Exam 2 Section A.
Q1.The rest energy of an object of mass m is given by:
1mark
Need a hint?
This is the most famous equation in physics. The speed of light appears squared.
Show worked solution
The rest energy is E0=mc2. The speed of light is squared, which is why even a tiny mass is equivalent to a huge amount of energy. The factor of one half belongs to the low speed kinetic energy formula, not to rest energy.
Q2.A small mass of 2.0×10−3 kg is completely converted to energy. Using c=3.0×108 m/s, the energy released is:
1mark
Need a hint?
Use E0=mc2 with c2=9.0×1016.
Show worked solution
E0=mc2=2.0×10−3×9.0×1016=1.8×1014 J. Option B comes from forgetting to square the speed of light, which throws the answer out by a factor of c.
Q3.A particle has rest energy E0 and moves at a speed where γ=3.0. Its total energy is:
1mark
Need a hint?
Total energy is Etot=γmc2=γE0.
Show worked solution
Total energy is Etot=γE0=3.0E0. Option B, 2.0E0, is the kinetic energy, since Ek=(γ−1)E0=2.0E0. Mixing up total energy with kinetic energy is the classic trap here.
Q4.A particle has a rest energy of 5.0×10−14 J. When moving, its total energy is 9.0×10−14 J. Its kinetic energy is:
1mark
Need a hint?
Kinetic energy is total energy minus rest energy.
Show worked solution
Ek=Etot−E0=9.0×10−14−5.0×10−14=4.0×10−14 J. Option B adds the two energies instead of subtracting, and option D forgets to take the rest energy out at all.
Q5.A proton has a mass of 1.7×10−27 kg. Taking c=3.0×108 m/s, calculate its rest energy. Show your working.
3marks
Work this on paper. The worked solution appears once you submit.
Show worked solution
Start from the rest energy equation and square the speed of light:
c2=(3.0×108)2=9.0×1016 m2/s2.
Then multiply by the mass:
E0=mc2=1.7×10−27×9.0×1016=1.5×10−10 J.
Frequently asked questions
What does mass-energy equivalence actually mean?
It means mass and energy are two forms of the same thing. A given amount of mass is equivalent to a fixed amount of energy, set by E equals mc squared, and one can be converted into the other.
Why is the energy from a tiny mass so large?
Because the mass is multiplied by the speed of light squared, and the speed of light is about 300 million metres per second. Squaring it gives a factor of 9 followed by 16 zeros, so even a gram of mass holds an enormous amount of energy.
What is the difference between total energy and kinetic energy?
Total energy is everything the object has, the rest energy plus the kinetic energy. Kinetic energy is only the extra energy due to motion, which is the total energy with the rest energy taken back out.
