IB Physics HLTopic 1 — Motion, Forces & EnergyPaper 1 & 2Work, Energy & Power~9 min read
Energy Flow (Sankey) Diagrams
Once you know that energy in equals useful energy out plus wasted energy, the obvious next question is: how much goes each way? A Sankey diagram answers that at a glance. It’s a flow picture where the width of each arrow is drawn in proportion to the amount of energy it carries — a fat arrow for a big flow, a thin one for a trickle. One look and you can see whether a device is putting most of its energy to good use or throwing most of it away as heat.
📘 What you need to know
A Sankey diagram is a scaled picture of an energy transfer
The width of each arrow is proportional to the amount of energy it represents
The arrow pointing right is the useful energy output (the desired store)
Arrows pointing down represent wasted energy (usually dissipated as heat)
Because energy is conserved: total energy in = useful energy out + wasted energy
A more efficient device has a wider useful arrow and a narrower wasted arrow
Reading a Sankey diagram
Every Sankey diagram starts with a single arrow on the left — the total energy (or power) going in. As that energy is transferred, the arrow splits. The part that ends up where you want it carries straight on to the right; the part that’s wasted peels away, usually downward. The clever bit is the scaling:
The golden rule of Sankey diagrams
Width of arrow ∝ amount of energy it carries
So an arrow carrying twice as much energy is drawn twice as wide. Because no energy is created or destroyed, the widths have to balance — the useful and wasted arrows together must be exactly as wide as the input arrow.
Conservation of energy on a Sankey diagram
Total energy in = Useful energy out + Wasted energy
A Sankey diagram for a device that’s 80% efficient. The 100 J input splits into an 80 J useful flow (wide) and a 20 J wasted flow (narrow). The two output widths add up to the input width — energy is conserved.
The width is the number. That’s the whole point of a Sankey diagram — you don’t have to read a single figure to see that this device wastes only a little energy, because the downward arrow is skinny. If you ever draw one in an exam, get the widths roughly to scale: a “wasted” arrow that’s fatter than the useful one is telling a lie about the physics.
Efficient vs inefficient: same input, different picture
Because the arrow widths are scaled, two devices doing the same job can produce completely different-looking Sankey diagrams. Compare a modern LED light bulb with an old-fashioned filament bulb. Both take in the same electrical energy, but they split it very differently:
Same 100 J in, very different pictures. The modern bulb sends most energy usefully to light (wide green arrow); the old filament bulb dumps almost all of it as heat (wide blue arrow), leaving a sliver as light. The narrower the wasted arrow, the more efficient the device.
WE 1
An electric heater is supplied with 2000 J of electrical energy and usefully transfers 1700 J to the room as thermal energy. Draw the energy accounting: how much energy is wasted?
Step 1 — energy is conserved
Total in = useful out + wasted
Step 2 — rearrange for wasted
wasted = total in − useful out
Step 3 — substitutewasted = 2000 − 1700Wasted energy = 300 JOn a Sankey diagram the useful arrow would be about 6 times wider than the wasted one (1700 vs 300).
When more than one thing is wasted
Some devices waste energy in several ways at once — a motor might lose energy to friction, to sound, and to heating the wires. On a Sankey diagram each of these gets its own downward arrow, each drawn to its own width. They still all have to add up to the input.
Input 240 J
splits into →
Useful 150 J
+
Sound 18 J
+
Heat 72 J
WE 2
An electric motor is supplied with 240 J of energy. It does 150 J of useful work lifting a load, and 18 J is transferred as sound. The rest is dissipated as heat. How much energy is wasted as heat?
Step 1 — the total must split into all its parts
Total in = useful + sound + heat
Step 2 — rearrange for the heat term
heat = total in − (useful + sound)
Step 3 — substituteheat = 240 − (150 + 18) = 240 − 168Heat wasted = 72 JCheck: 150 + 18 + 72 = 240 ✓ — the arrows balance.
Turning widths back into numbers
Sankey diagrams work both ways. If you’re told the scale — how much energy one arrow’s width stands for — you can measure any arrow and read off its energy, or the other way round: work out how wide an arrow should be for a given energy.
WE 3
On a Sankey diagram the input arrow is 8.0 cm wide and represents 400 J. If the useful output is 260 J, how wide should the useful output arrow be drawn?
Step 1 — width is proportional to energy
width = input width × (energy ÷ input energy)
Step 2 — substitutewidth = 8.0 × (260 ÷ 400)width = 8.0 × 0.65Useful arrow = 5.2 cm wideThe wasted 140 J would need an arrow 8.0 × (140÷400) = 2.8 cm wide. Check: 5.2 + 2.8 = 8.0 cm ✓
🛠️ Drawing a Sankey diagram
Plan the widths first. Decide how wide the input arrow is, then how wide the useful and wasted arrows must be to match their energies.
Draw the input arrow on the left, with the straight line running across the top.
Add the useful arrow continuing to the right, making sure its width is correct.
Mark the start and end of each wasted arrow peeling downward — check the distance apart is right for its energy.
Join the markings to finish each wasted arrow. Double-check the output widths add up to the input width.
💡 Top tips
Widths must add up. The useful and wasted arrow widths together always equal the input width — it’s conservation of energy in picture form.
Right = useful, down = wasted. Keep to the convention so your diagram reads correctly.
Get the scale right. If 1 cm stands for a fixed number of joules, use it consistently for every arrow.
Sankey diagrams work for power too — just label the arrows in watts instead of joules; the rules are identical.
Quick recap: A Sankey diagram shows an energy transfer with arrow widths drawn in proportion to the energy carried. The useful output goes right, wasted energy goes down, and the output widths add up to the input width because energy is conserved (total in = useful out + wasted). A more efficient device has a wider useful arrow and a thinner wasted one.
⚠ Common mistakes
Drawing arrows not to scale — the whole value of a Sankey diagram is that width represents energy
Making the output widths not add up to the input width, which breaks conservation of energy
Confusing useful and wasted — which is useful depends on what the device is meant to do
Forgetting that a device can have several wasted arrows, not just one
Thinking wasted energy has disappeared — it’s dissipated (usually as heat), not destroyed
You’ve now seen the same idea twice: total in = useful out + wasted. A Sankey diagram just draws it. The natural next step is to put a number on how good a device is — that’s efficiency, the fraction of the input that comes out useful. Once you can read a Sankey diagram, the efficiency formula will feel like it’s just measuring the width of the useful arrow.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
