SPH4U Physics: the course map (25 ideas across strands B to F), derive-before-memorize for the kinematics and energy chains, and how to study it

SPH4U is Ontario's Grade 12 university physics: dynamics in two dimensions, energy and momentum, gravitational, electric and magnetic fields, the wave nature of light, and modern physics (relativity and the quantum). Our map has 25 ideas in teaching order across strands B to F. The way to study it is to derive each equation once from the two or three you know, so on the test you choose the model before the formula. Here is the map and the plan.

What SPH4U is, and what it assumes

SPH4U is the Ontario course code for Physics, Grade 12, University Preparation. Its prerequisite is SPH3U, the Grade 11 university physics course, and almost everything in SPH4U picks up a Grade 11 idea and runs it further: motion in a line becomes motion in a plane, one force becomes a free-body diagram with four, energy stays conserved but now trades with momentum.

The mathematics it leans on is Grade 11 and 12 algebra and trigonometry. You resolve vectors into components with sine and cosine, solve quadratics, and read slopes and areas off graphs. The derivative idea from MCV4U is useful informally, because "the slope of the velocity graph" is a derivative in disguise, but the course is written to be done without calculus. A student taking SPH4U without MCV4U is not at a disadvantage in SPH4U itself.

What it leads to is first-year physics and the first year of any engineering program, and the gap between SPH4U and those courses is mostly calculus, not physics. The ideas carry; the tools get sharper.

One note on the shape of the course. The curriculum's strand A covers the skills of scientific inquiry, which run through every unit rather than standing on their own, which is why our map starts at strand B. The Ministry's own course description is the source for the code, title and prerequisite; the descriptions below are ours.

The course map: 25 ideas in the order they are taught

These are the 25 ideas of our SPH4U map, in the order most classes meet them, grouped by strand. Chapter 3 of the free playbook is the method for turning a list like this into a one-page map with a mark beside each idea; this list is what you would write on it.

Dynamics (strand B)

1. Two-dimensional kinematics and projectile motion
2. Newton's laws; free-body diagrams; net force
3. Friction and motion on inclined planes
4. Uniform circular motion and centripetal force
5. Relative motion in two dimensions

Energy and momentum (strand C)

6. Work and kinetic energy
7. Conservation of mechanical energy
8. Power and efficiency
9. Impulse and momentum
10. Conservation of momentum in one dimension; inelastic collisions
11. Elastic collisions
12. Momentum in two dimensions

Gravitational, electric and magnetic fields (strand D)

13. Newton's law of universal gravitation
14. Orbital motion and satellites
15. Gravitational potential energy and potential
16. Coulomb's law and electric fields
17. Electric potential and potential difference
18. Magnetic fields and the force on moving charges and currents

The wave nature of light (strand E)

19. Interference of light; Young's double slit
20. Diffraction and diffraction gratings
21. Polarization of light

Revolutions in modern physics (strand F)

22. Special relativity: time dilation and length contraction
23. Mass–energy equivalence
24. The photoelectric effect and photon energy
25. Energy levels and matter waves

Twelve of the 25 are in the first two strands, and those twelve are where the marks and the difficulty are concentrated, because they chain. Idea 7 needs idea 6, which needs idea 1. That chaining is the reason for the next section.

Derive before you memorize: the two chains

A formula sheet for SPH4U runs to thirty or forty equations. Nobody remembers forty equations under exam pressure, and nobody needs to, because almost all of them are two or three equations rearranged. If you can rebuild an equation from its parents in a minute, you own it; if you can only recognise it on the sheet, you will reach for the wrong one when the problem is dressed up. Fiorella and Mayer (2016) review the evidence that learning is stronger when students generate the material themselves, by explaining, drawing or deriving, than when they receive it finished, and Chi and colleagues (1994) showed the same with physics text: students prompted to explain each step to themselves understood more than students who read the text twice. Deriving is self-explanation with algebra.

Two chains carry most of the course.

The kinematics chain

Start from two equations you can picture. A constant acceleration means velocity changes at a steady rate, so v = v₀ + at. Displacement is the area under the velocity graph, which for a straight line from v₀ to v is the average velocity times the time, so Δd = v₀t + ½at².

Now eliminate t. From the first, t = (v − v₀)/a. Substitute into the second and simplify, and you get v² = v₀² + 2aΔd. Do that elimination once, on paper, slowly. After that the third equation is not a fact to remember but a thing you did.

Projectile motion is the same two equations run twice, once in each direction. Horizontally there is no force, so a = 0 and x = v₀ₓt. Vertically a = −g, with g = 9.8 m/s² kept exact in the working, as our practice bank keeps it. The two directions share only the time.

Worked: a ball rolls off a ledge 20 m high at 15 m/s. How far from the base does it land?

• Vertical first, because that fixes the time. Starting with no vertical velocity, 20 = ½(9.8)t², so t² = 2 × 20/9.8 = 4.08 and t = 2.02 s.
• Horizontal second. Range = 15 × 2.02 = 30.3 m.

The common wrong line is using 15 m/s somewhere in the vertical equation. The horizontal speed has nothing to do with how long the fall takes; a ball dropped from rest beside it lands at the same instant.

The energy–momentum chain

Work is force times displacement along the force: W = FΔd. Put Newton's second law in for F and the kinematics chain in for Δd, and the work done on an object comes out as the change in ½mv². That is where kinetic energy comes from; it is not a separate fact. Lifting an object against gravity at steady speed does work mgΔh, which is where gravitational potential energy comes from. When no friction acts, the total of the two stays fixed: conservation of mechanical energy.

Momentum comes from the same second law written the other way. F = ma = m(Δv/Δt), so FΔt = mΔv = Δp: impulse equals the change in momentum. Newton's third law says two colliding objects feel equal and opposite forces for the same time, so their impulses cancel and total momentum is conserved. Every collision problem in strand C is that sentence.

Worked: a 2.0 kg cart moving at 3.0 m/s hits a 1.0 kg cart at rest and the two stick together. What is their speed afterwards, and how much kinetic energy was lost?

• Momentum before = 2.0 × 3.0 + 1.0 × 0 = 6.0 kg·m/s. After, one 3.0 kg object at speed v, so 3.0v = 6.0 and v = 2.0 m/s.
• Kinetic energy before = ½ × 2.0 × 3.0² = ½ × 2.0 × 9.0 = 9.0 J.
• Kinetic energy after = ½ × 3.0 × 2.0² = ½ × 3.0 × 4.0 = 6.0 J.
• Lost: 9.0 − 6.0 = 3.0 J, to sound, heat and the deformation of the carts.

The lesson in the numbers: an inelastic collision conserves momentum and does not conserve kinetic energy. Students who set this problem up with energy conservation get v = 2.45 m/s, which is wrong, and the error is a model choice, not arithmetic. Name the model before the formula: "they stick, so momentum."

The fields strand: one law, three times

Strand D looks like three topics and is one shape repeated. Newton's law of universal gravitation says the force between two masses falls off with the square of the distance: F = Gm₁m₂/r². Coulomb's law says the same of two charges: F = kq₁q₂/r². Learn the first properly and the second is a change of letters. A field is that force per unit of the thing that feels it, so g = F/m near a mass and E = F/q near a charge. Potential is energy per unit of that same thing: gravitational potential is energy per kilogram, electric potential is energy per coulomb, and potential difference, the volt, is a difference in energy per coulomb between two points.

Orbital motion joins strand B to strand D: the gravitational force is the centripetal force, so Gm₁m₂/r² = mv²/r, and from that one line come orbital speed and period. The magnetic force on a moving charge is the one idea in the strand with a new shape, F = qvB sin θ, and a direction found with the right hand.

Here is the symbol rule, which is our practice bank's rule and the examiner's: keep G, k, h and c as letters until the last line, and keep g as 9.8 exactly. Substituting 6.67 × 10⁻¹¹ in the first line produces a page of exponents, and a page of exponents is where the sign error hides. Solve the algebra with letters, then put the numbers in once.

Worked, from strand B with the same habit: a 1,200 kg car takes a curve of radius 50 m at 20 m/s. What centripetal force does the road supply?

• The model: uniform circular motion, so F = mv²/r.
• F = 1,200 × 20²/50 = 1,200 × 400/50 = 9,600 N.

Keep the letters until the last line and the units come out with the answer: kg × (m/s)² ÷ m = kg·m/s² = N.

Light and modern physics: vocabulary plus one calculation each

Strands E and F are different in kind. Fewer equations, more models, more words, and a kind of question that asks you to explain what is observed and why one model fits it. For each of the seven ideas, hold three things: what is observed, which model explains it, and the one equation. Then own one calculation type per idea.

• Young's double slit: bright and dark fringes on a screen; light as a wave, with interference; the fringe spacing from the wavelength, the slit separation and the distance to the screen.
• Diffraction and gratings: light bends around an edge and a grating throws sharp spectra; the same wave model; the grating equation for the angle of each order.
• Polarization: a filter cuts the intensity; light as a transverse wave; the direction of the field that passes.
• Special relativity: moving clocks run slow and moving rulers shrink; the two postulates; a time dilation calculation with the factor γ (at 0.60c, γ = 1/√(1 − 0.36) = 1.25).
• Mass–energy equivalence: E = mc², with c kept as a letter until the last line.
• The photoelectric effect: electrons leave a metal only above a threshold frequency, whatever the brightness; light as photons of energy E = hf; a photon-energy or stopping-voltage calculation.
• Energy levels and matter waves: atoms emit only certain colours and electrons diffract; quantised energy levels and the de Broglie wavelength.

Study these the way you would study a biology unit: recall cards for the observations and the models, plus one worked problem per idea redone cold. The science study guide how to study for science has the recall-plus-one-problem routine in full.

How to study SPH4U

Grade 12 physics is a course of set-ups. In our experience of marking, most lost marks are a wrong free-body diagram, a dropped sign on a component, or the wrong model chosen, not a slip on the calculator. The method follows from that.

1. Derive each equation once in your own notebook. When a new equation appears in class, close the book that evening and rebuild it from its parents. The two chains above are the template; the fields strand repeats them. If you cannot rebuild it, you do not have it yet, and that is the thing to ask about tomorrow.
2. Name the model before the formula, on every problem. Write one word at the top before anything else: "kinematics", "forces", "energy" or "momentum". If the word is wrong, every line below it is wrong, and this is the habit that catches the cart problem above. The physics section of the science study guide calls this the first line of every solution.
3. Mixed sets across strands B and C from about week 6. Once energy arrives, a set of ten problems should shuffle forces, projectiles, energy and momentum, because the exam will. Why mixed practice beats blocked practice has the evidence and how to shuffle an exercise set by hand.
4. An error log for set-ups. Every wrong line goes in a list with the right line beside it, and most of the entries will say the same few things: "forgot the normal force on the incline", "sign of g", "used energy where momentum". Redo the problem cold the next day. How to study math has the worked-example cycle this follows: read one solved problem, redo it cold, do a sibling with new numbers.
5. Recall cards for the modern-physics vocabulary. The three-things rule above, one card per thing: observation on the front, model on the back. How to write a flashcard keeps the answer off the front.

On physics homework help: a worked solution read after your own attempt is help, and a solution read instead of an attempt is a night of rereading. The error log only works on your own wrong lines.

A week-by-week plan for a semester: the SPH4U exam review that starts in week 1

An 18-week semester, the strands in the order most classes teach them. Each row has the derivation to redo cold that week and the problem set to do mixed. This is the exam review done as you go; the last two weeks only collect it.

Weeks	Strand and ideas	Derivation to redo cold	Mixed set
1–4	Dynamics, ideas 1–5	v² = v₀² + 2aΔd from the two parent equations; a projectile in x and y	Projectiles and forces together from week 3; add inclines and circles as they arrive
5–8	Energy and momentum, ideas 6–12	Kinetic energy from work; impulse from Newton's second law; momentum conservation from the third	From week 6, ten problems shuffled across strands B and C, every week
9–12	Fields, ideas 13–18	Orbital speed from gravity as the centripetal force; potential as energy per kilogram, then per coulomb	Fields with the strand B and C set, two problems each
13–14	Light, ideas 19–21	Fringe spacing from the path difference	Recall cards for the observations and models, one problem each
15–16	Modern physics, ideas 22–25	γ from the two postulates; E = hf against the threshold frequency	Cards plus one calculation per idea; a mixed set over strands D to F
17	Mixed review across all five strands	Every derivation in the table, from a blank page	A full timed set of twenty problems, all strands, marked and logged
18	Exam week	None new; read the error log	The light review the night before, and sleep

The week-18 row is the one-week plan in how to study for a test in one week: a diagnostic on day 7, spaced sessions, a timed run two days out, a light review the night before.

What carries into first year

Almost all of it, with sharper tools. The two chains come back in first-year physics with calculus in them: "the slope of the velocity graph" becomes the derivative, "the area under the force graph" becomes an integral, and the kinematics equations fall out of the derivative in two lines. The MCV4U page explains the derivative idea that does that work. The fields strand returns as a full course in electromagnetism, with the same inverse-square law at the front. The modern strand returns as a whole course of its own, usually in second year. A student who derived in SPH4U rather than memorizing finds first-year physics a change of notation, not a change of subject.

SPH4U on one screen

• Five strands, 25 ideas: dynamics (5), energy and momentum (7), fields (6), light (3), modern physics (4). Strand A's inquiry skills run through all of them.
• Two chains carry the marks: kinematics (two parent equations, eliminate t; projectiles in x and y) and energy–momentum (work gives kinetic energy; the second law gives impulse; the third law gives conservation).
• The symbol rule: G, k, h and c stay as letters until the last line; g = 9.8 exactly.
• Name the model first: one word at the top of every solution.
• Light and modern physics: observation, model, one equation; recall cards plus one worked problem per idea.
• Mixed sets from week 6; an error log of set-ups; derive every equation once.