IB Physics HLTopic 1 — Motion, Forces & EnergyPaper 1 & 2Work, Energy & Power~11 min read
Conservation of Mechanical Energy
You’ve now met all three moving-and-storing energies: kinetic, gravitational potential, and elastic potential. Add them up and you get an object’s mechanical energy. The powerful idea in this page is that, as long as no friction or drag is stealing energy away, that total stays exactly the same — it just keeps swapping between the three forms. A pendulum, a rollercoaster, a bouncing spring: the mechanical energy is constant, so what one store loses, another gains. That single principle solves a huge range of problems without any need for forces or acceleration.
📘 What you need to know
Mechanical energy is the sum of kinetic, gravitational potential and elastic potential energy
Mechanical energy = Ek + ΔEp + EH
With no resistive forces, mechanical energy is conserved — it stays constant
Energy just transfers between the forms: loss in one = gain in another
For a falling object: loss in GPE = gain in KE
For a horizontal spring: loss in elastic PE = gain in KE
A non-conservative force like friction removes mechanical energy (usually as heat), so the total drops
What is mechanical energy?
Mechanical energy is simply the running total of an object’s kinetic and potential energies:
Mechanical energy
Mechanical energy = Ek + ΔEp + EH
The classic example is a mass bouncing on a spring, which cycles through all three. As it moves it constantly converts energy between kinetic (moving), gravitational potential (height) and elastic potential (spring stretch or squash). The total stays put — only the split changes.
A mass on a vertical spring. At the top (A) it’s momentarily still and high: GPE is greatest. Passing through the middle (B) it moves fastest: KE is greatest. At the lowest point (C) the spring is stretched most: EPE is greatest. The total never changes.
Energy at each position
Tracking the three stores through a full bounce shows how they trade off — while one is at a maximum, another is at zero, but they always sum to the same total:
Position
GPE
KE
EPE
A (top)
Maximum
Zero
Some
B (middle)
Some
Maximum
Some
C (bottom)
Minimum
Zero
Maximum
For a horizontal mass on a spring, the height never changes, so there’s no gravitational potential energy to worry about — the spring only swaps energy between kinetic and elastic potential.
The principle of conservation of mechanical energy
When there are no resistive forces — no friction, no air resistance — the total mechanical energy of a system doesn’t change:
Conservation of mechanical energy
Total mechanical energy = constant (when no resistive forces act)
This is really just conservation of energy applied to the mechanical stores. Since the total is fixed, energy lost by one store must be gained by another. For an object falling or a pendulum swinging down:
Falling or swinging
Loss in gravitational potential energy = Gain in kinetic energy
And for a mass on a horizontal spring:
Horizontal spring
Loss in elastic potential energy = Gain in kinetic energy
As an object falls, gravitational potential energy (purple) converts into kinetic energy (blue). Each bar is the same total height — the mechanical energy is conserved, only its split changes.
WE 1
A 0.30 kg ball is released from rest and falls 1.2 m. Ignoring air resistance, use conservation of mechanical energy to find its speed at the bottom. (Take g = 9.81 m s−2.)
Step 1 — loss in GPE = gain in KE
mgΔh = ½mv²
Step 2 — mass cancels, rearrange for v
v = √(2gΔh)
Step 3 — substitutev = √(2 × 9.81 × 1.2) = √23.5v = 4.9 m s⁻¹ (2 s.f.)No forces or times needed — energy conservation gets there in one step, and the mass cancels.
When friction is present
Real systems lose mechanical energy to non-conservative forces like friction and air resistance. These dissipate energy as heat and sound, so the total mechanical energy drops. The change in mechanical energy equals the work done by those resistive forces. In these problems you account for the fraction lost:
Start energy (e.g. GPE)
− energy lost to friction →
Useful KE at the end
WE 2
A 500 kg rollercoaster car starts from rest at the top of a drop 40 m high. Friction dissipates 20% of the initial gravitational potential energy. Find the car’s speed at the bottom. (Take g = 9.81 m s−2.)
Step 1 — only 80% of the GPE becomes KE
½mv² = 0.80 × mgΔh
Step 2 — mass cancels, rearrange for v
v = √(0.80 × 2gΔh)
Step 3 — substitutev = √(0.80 × 2 × 9.81 × 40) = √628v = 25 m s⁻¹ (2 s.f.)The missing 20% has become heat in the wheels and track — still conserved overall, just not as mechanical energy.
“Work done” in an exam question usually means “how much energy was transferred”. The trick is to account for all the energy: what started as one store, what ended as another, and what leaked away to friction. Because total energy is always conserved, those three always balance — even when the mechanical energy alone doesn’t.
WE 3
A 0.40 kg block sits against a horizontal spring of spring constant 250 N m−1, compressed by 0.12 m. The spring is released on a frictionless surface. Find the speed of the block as it leaves the spring.
Step 1 — horizontal spring: loss in EPE = gain in KE
½kΔx² = ½mv²
Step 2 — find the elastic PE storedEH = ½ × 250 × 0.12² = 1.8 JStep 3 — set equal to KE and solve for vv = √(2 × 1.8 ÷ 0.40) = √9.0v = 3.0 m s⁻¹No height change on a horizontal surface, so GPE plays no part — only elastic PE and KE.
🛠️ Solving a conservation of energy problem
Identify the start and end states — which stores are full at each?
Write “energy lost by one store = energy gained by another” (e.g. GPE → KE).
If resistive forces act, include the fraction lost, so only the remaining part transfers.
Substitute the energy equations (mgΔh, ½mv2, ½kΔx2) and cancel any common factors like mass.
Rearrange and solve, taking the square root last if you’re finding a speed.
💡 Top tips
Mass often cancels. In pure GPE-to-KE problems, mass drops out — you don’t always need it.
Height is vertical. On a slope, use the vertical drop (Δh = length × sin θ), not the distance along it.
Account for losses. If a percentage is “not transferred”, only the rest becomes useful energy — multiply the start energy by the fraction that does transfer.
Horizontal springs skip GPE. No height change means no gravitational potential term.
Quick recap: Mechanical energy is the sum of kinetic, gravitational potential and elastic potential energy. With no resistive forces it’s conserved, so a loss in one store equals a gain in another (GPE → KE for a fall, EPE → KE for a spring). Friction and drag are non-conservative — they remove mechanical energy as heat, so you must account for the fraction lost.
⚠ Common mistakes
Ignoring friction losses when a question states energy is dissipated or “not transferred”
Using the slope distance instead of the vertical height for GPE
Forgetting that mass cancels in GPE-to-KE problems, then panicking when it isn’t given
Including gravitational PE for a horizontal spring, where the height doesn’t change
Thinking friction destroys energy — it converts mechanical energy to heat, so total energy is still conserved
That’s the whole energy toolkit assembled: the three stores, and the rule that ties them together. Next we shift from how much energy is transferred to how quickly — that rate is power, and it’s the natural companion to everything you’ve just learned about work and energy.
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