🎯 Hook – the runner who goes nowhere
A runner completes one lap of a 400 m track in 80 s. Ask "how fast was she going?" and there are two correct but very different answers. Her average speed was \(400/80 = 5.0\ \text{m s}^{-1}\) — but her average velocity was exactly zero, because she ended up back where she started. Distance and displacement answer different questions, and the gap between them is the whole reason physics separates scalars from vectors.
📏 Definitions and common quantities
A scalar is a quantity with magnitude (size) only; scalars add by ordinary arithmetic (3 kg + 4 kg = 7 kg). A vector has both magnitude and direction; vectors add by vector rules (Lesson 03): 3 N east + 4 N north = 5 N, not 7 N.
| Scalars | Vectors |
|---|---|
| Distance, speed, time | Displacement, velocity, acceleration |
| Mass, density | Weight, gravitational field strength |
| Energy, work, power | Force, momentum, impulse |
| Temperature, pressure | Electric field strength, magnetic flux density |
| Charge, current, p.d., resistance | — |
Equal vectors have the same magnitude and the same direction, wherever they are drawn. The negative of a vector has the same magnitude but the opposite direction; multiplying a vector by a scalar \(k\) scales its length by \(k\) (and reverses it if \(k\) is negative).
↔️ Sign conventions, distance vs displacement
Sign conventions in one dimension: choose a positive direction (up or right, say) and keep it for the whole problem. A velocity of \(-5\ \text{m s}^{-1}\) means 5 m s⁻¹ in the negative direction; a speed is never negative.
Distance is the total length of path travelled (scalar). Displacement is the straight-line change in position from start to finish, with direction (vector). Average speed = total distance/time; average velocity = displacement/time.
🌀 Deeper notes — signed scalars, circular motion, frames
Signed scalars are not vectors. Charge (−e), work (negative when a force opposes motion), temperature (−5 °C) and potential (−3 V) have signs, but the sign shows type or relative value, not a spatial direction — they add arithmetically.
Why is current a scalar? Current has a direction along a wire, but currents at a junction combine by arithmetic (Kirchhoff's first law), not by vector addition — so IB physics treats it as a scalar.
Constant speed can still mean acceleration. Velocity is a vector, so changing direction changes velocity. An object in uniform circular motion accelerates towards the centre even though its speed is constant.
Frames of reference. Velocity depends on the observer: a passenger walking at 1 m s⁻¹ forwards in a train moving at 30 m s⁻¹ has a velocity of 31 m s⁻¹ relative to the ground. Relative velocity is a vector subtraction: \(v_{AB}=v_A-v_B\).
📊 Graph mastery — motion of a ball thrown up and caught
| Graph type | Axes (X, Y) | Shape | What to read |
|---|---|---|---|
| Distance–time | t · distance | Always rising | Gradient = speed (never negative) |
| Displacement–time | t · displacement | Rises then falls back to zero | Gradient = velocity; sign shows direction |
| Velocity–time | t · v | Straight line crossing zero at the top | Sign change shows the reversal of direction |
| Speed–time | t · speed | V-shape touching zero at the top | Always ≥ 0, mirrors |v| |
✍️ Worked example 1 — walk and lap
✍️ Worked example 2 — change of velocity, not speed
🌍 Real-world: relative velocity in traffic
Car A moves east at 25 m s⁻¹, car B east at 18 m s⁻¹. Velocity of A relative to B = \(25-18=7\ \text{m s}^{-1}\) east. If B instead moved west at 18 m s⁻¹: \(25-(-18)=43\ \text{m s}^{-1}\) east — the closing speed in a head-on situation is far larger than either car's speed alone.
🚀 Going further — products of vectors
Multiplying two vectors can give a scalar: work \(W=Fs\cos\theta\) (Theme A). Or it can give a vector: torque and magnetic force come from a different kind of vector product (Theme D). The dot/cross distinction previews material you will formalise later.
Challenge: a force of 10 N acts at 60° to a 4.0 m displacement. What scalar does \(Fs\cos\theta\) give here? Work done = \(10\times4.0\times\cos60^\circ = 20\ \text{J}\).
📋 Quick practice — answers in italics
(a) Classify as scalar or vector: kinetic energy, momentum, pressure, impulse. Scalar, vector, scalar, vector.
(b) A cyclist rides 6.0 km north then 8.0 km south. Find distance and displacement. Distance = 14.0 km; displacement = 2.0 km south.
(c) A ball falls and hits the floor at 6.0 m s⁻¹, rebounding at 4.0 m s⁻¹ (up positive). Calculate the change in velocity. \(\Delta v = (+4.0)-(-6.0)=+10\ \text{m s}^{-1}\).
(d) Explain why "the speed was −15 m s⁻¹" is incorrect. Speed is a scalar and is never negative — only velocity can carry a sign.
(e) Give an example of a quantity that can be negative but is a scalar. Temperature (−5 °C), or electric charge (−e), or work done against a force — the sign shows type, not spatial direction.
⚠️ Trap corner – common mistakes
| Misconception | Correct view |
|---|---|
| Speed can be negative | Speed is never negative; velocity can be, since it carries direction |
| Average velocity = average speed | Only equal for motion in one straight line without reversing; a closed loop gives average velocity 0 |
| Mass and weight are the same quantity | Mass (scalar, kg) and weight (vector force, N) are different quantities with different units |
| A negative sign on a scalar means a direction | Signs on charge, work or temperature show type or relative value, not a spatial direction |
| Equal magnitude means equal vectors | 5 N north ≠ 5 N east — direction matters as much as size |
| Constant speed means constant velocity | Constant speed in a circle is not constant velocity — direction is always changing |
Complete the statements:
✓ (b) Scalar — its sign shows the type of charge, not a spatial direction, and charges combine by arithmetic.
✓ (c) Work (scalar) and torque (vector) are a valid pair — both measured in N m (or J).
✓ (b) Displacement \(=\sqrt{5.0^2+12.0^2}=13.0\) km; bearing \(=\tan^{-1}(12.0/5.0)=\mathbf{067^\circ}\).
✓ (c) Average velocity \(=13.0/1.5=\mathbf{8.7\ km\,h^{-1}}\) on bearing 067°.
✓ (b) After half a revolution the velocity has reversed: \(\Delta v = v_2-v_1 = 4.0-(-4.0) = \mathbf{8.0\ m\,s^{-1}}\).
✓ (c) Directly opposite to the initial velocity (anti-parallel to it).
| t/s | x/m | y/m |
|---|---|---|
| 0 | 0.00 | 0.00 |
| 1 | 2.00 | 0.00 |
| 2 | 4.00 | 0.00 |
| 3 | 4.00 | 1.50 |
| 4 | 4.00 | 3.00 |
| 5 | 2.00 | 3.00 |
| 6 | 0.00 | 3.00 |
✓ (b) Displacement \(=(0,3.00)\) m, i.e. 3.00 m north; average velocity \(=3.00/6=\mathbf{0.50\ m\,s^{-1}}\) north.
✓ (c) t = 2–4 s: \(\Delta y=3.0\) m in 2 s → 1.5 m s⁻¹ north. t = 4–6 s: \(\Delta x=-4.0\) m in 2 s → 2.0 m s⁻¹ west.
✓ (d) Each coordinate carries ±0.05 m, so a component difference carries \(\sqrt{0.05^2+0.05^2}\approx\pm0.07\) m — small (~2%) next to the 3.00 m displacement, so not significant.
✓ (e) Increase the camera frame rate / use higher-resolution tracking or calibrated reference markers to reduce the ±0.05 m uncertainty.
✓ Example: 3 N east and 4 N north add arithmetically to "7 N" if treated as scalars, but the true (vector) resultant is 5 N at 53° north of east — a different magnitude and a missing direction.
✓ Against: at a junction, currents combine by simple arithmetic (Kirchhoff's first law) rather than by vector addition rules — a true vector quantity would need components and angles to combine.
✓ IB convention: because it does not obey vector addition, current is treated as a scalar at this level (current density, a genuine vector, appears only in more advanced treatments).
✓ The sign only shows the direction of energy transfer, not a direction in space — work remains a scalar (a transferred amount of energy), unlike force or displacement.
✓ Since every term flips together, the final physical conclusion — which direction something actually moves, and by how much — is identical either way; only the algebra's signs change, not the physics.
✓ Speed is the magnitude of velocity, so it inherits the same frame-dependence: the same passenger's speed is 1 m s⁻¹ or 31 m s⁻¹ depending on which frame is used.