IB Physics HLTopic 1 — Motion, Forces & EnergyPaper 1 & 2Work, Energy & Power~10 min read
Kinetic Energy
Anything that’s moving carries energy simply because it’s moving — that’s kinetic energy. A rolling ball, a speeding car, a falling raindrop: they all have it, and the faster and heavier they are, the more they’ve got. The important twist, and the thing exams love to test, is that speed matters far more than you’d expect. Double the speed and the kinetic energy doesn’t double — it quadruples. That single fact explains everything from why braking distances balloon at high speed to why a small increase in a car’s speed makes a crash so much worse.
📘 What you need to know
Kinetic energy is the energy an object has because of its motion
It’s calculated with Ek = ½mv2
Kinetic energy depends on the square of the speed — double the speed, four times the energy
An object keeps its kinetic energy unless its speed or mass changes
A falling object gains kinetic energy as it loses gravitational potential energy
In terms of momentum p, kinetic energy can also be written Ek = p2 / 2m
Kinetic energy is a scalar — it’s never negative
The kinetic energy equation
For an object of mass m moving at speed v, the kinetic energy is:
Kinetic energyEk = ½mv2
where Ek is the kinetic energy in joules (J), m is the mass in kilograms (kg), and v is the speed in metres per second (m s−1). Notice carefully that only the speed is squared — not the mass, and not the ½. This is the single most common slip, so it’s worth burning in now.
WE 1
A cyclist and bike have a combined mass of 75 kg and are travelling at 8.0 m s−1. Calculate their kinetic energy.
Step 1 — write the equation
Ek = ½mv²
Step 2 — substitute (square the speed only)Ek = ½ × 75 × 8.0²Ek = ½ × 75 × 64Ek = 2400 JOnly the 8.0 gets squared — the mass and the half stay as they are.
Why speed matters so much
Because the speed is squared, kinetic energy climbs much faster than the speed itself. If you double the speed, you multiply the kinetic energy by 2² = 4. Triple the speed, and it’s 3² = 9 times as much. The graph of kinetic energy against speed isn’t a straight line — it’s a curve that gets steeper and steeper.
Kinetic energy against speed is a parabola. Because Ek ∝ v², quadrupling the speed multiplies the energy by sixteen — the curve rockets upward.
This is why speed limits matter so much for safety. A car at 30 m s−1 has four times the kinetic energy of the same car at 15 m s−1, even though it’s only going twice as fast — and all of that energy has to be got rid of by the brakes (or, in a crash, by crumpling metal). Square laws are sneaky like that.
Rearranging for speed or mass
The same equation works backwards. If you know the kinetic energy and want the speed, rearrange:
Speed from kinetic energyv = √(2Ek / m)
WE 2
A car of mass 1200 kg has 150 000 J of kinetic energy. Calculate its speed.
Step 1 — start from Ek = ½mv² and rearrange
v = √(2Ek ÷ m)
Step 2 — substitutev = √(2 × 150000 ÷ 1200)v = √250v = 15.8 m s⁻¹ (3 s.f.)Undo the squaring last — the square root is the final step, after the doubling and dividing.
Kinetic energy and momentum
Kinetic energy and momentum both depend on mass and velocity, so it’s no surprise they’re linked. Since momentum is p = mv, we can rewrite the kinetic energy in terms of momentum:
Kinetic energy in terms of momentumEk = p2 / 2m
These are the same equation in disguise — substituting p = mv into p2 / 2m gives you back ½mv2. The momentum form is especially handy in particle physics, where particles are usually described by their momentum rather than their speed.
Ek = p²/2m
put p = mv →
(mv)²/2m
simplify →
½mv²
WE 3
A proton of mass 1.67 × 10−27 kg has a momentum of 3.2 × 10−21 kg m s−1. Calculate its kinetic energy.
Step 1 — use the momentum form
Ek = p² ÷ 2m
Step 2 — substituteEk = (3.2 × 10⁻²¹)² ÷ (2 × 1.67 × 10⁻²⁷)Step 3 — evaluateEk = (1.024 × 10⁻⁴¹) ÷ (3.34 × 10⁻²⁷)Ek = 3.1 × 10⁻¹⁵ J (2 s.f.)The momentum form saves you from working out the speed first — ideal for tiny particles.
🛠️ Solving a kinetic energy problem
List what you have — mass, speed, momentum or energy?
Pick the right form: use ½mv2 when you know the speed, and p2 / 2m when you know the momentum.
Square the speed only — never the mass or the half.
Rearrange before substituting if you’re solving for speed or mass.
Undo the square last: take the square root at the very end when finding a speed.
💡 Top tips
Only the speed is squared. Not the mass, not the ½ — this is the number-one mistake in this topic.
Kinetic energy is a scalar. Even for a “loss” of kinetic energy, don’t attach a negative sign to the energy itself.
Both forms are on the data sheet. Use ½mv2 in mechanics and p2 / 2m in particle physics — they’re identical.
Think in squares. If a question changes the speed by a factor, the energy changes by that factor squared.
Quick recap: Kinetic energy is the energy of motion, Ek = ½mv2, measured in joules. Because the speed is squared, kinetic energy grows as the square of the speed — double the speed, four times the energy. It can also be written Ek = p2 / 2m using momentum, and it’s always a positive scalar.
⚠ Common mistakes
Squaring the mass or the ½ — only the speed is squared
Forgetting to square the speed altogether
Taking the square root too early when rearranging for speed — it comes last
Giving a negative kinetic energy — energy is a scalar and can’t be negative
Assuming doubling the speed doubles the energy — it quadruples it
Kinetic energy is the “moving” store. Its natural partner is the “height” store — the energy an object gains just by being lifted up. That’s gravitational potential energy, and it’s what we look at next. The two constantly swap back and forth in falling objects, pendulums and rollercoasters, so getting comfortable with both is the key to the whole section.
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