IB Maths AA SLTopic 5 — CalculusPaper 1 & 2~8 min read
Displacement, Velocity & Acceleration
Kinematics is the language of motion. Before you do any calculus, you need to nail down five quantities — displacement, distance, velocity, speed, acceleration — and the sign conventions that go with them. Mix up displacement and distance and you’ll lose marks. Get the conventions right and the rest is just careful arithmetic.
📘 What you need to know
Motion in this course is along a straight line only — horizontal (positive = right) or vertical (positive = up).
Displacement can be negative — it includes direction. Distance is always positive.
Velocity can be negative — it includes direction. Speed is always positive (it’s |v|).
If v and a have the same sign, the particle is speeding up. Different signs → slowing down.
On a velocity-time graph: gradient = acceleration, area = displacement (with sign) or distance (without).
“At rest” means v = 0. “Initially” means t = 0.
The five quantities
Every kinematics problem revolves around these. Memorising symbols, units and sign conventions saves you from the most common mistakes.
The five quantities — symbols, units, signs
Quantity
Symbol
Units
Sign
Displacement position relative to a fixed point
s
m
+ or −
Distance total path length travelled
d
m
always +
Velocity rate of change of displacement
v
m s⁻¹
+ or −
Speed magnitude of velocity
|v|
m s⁻¹
always +
Acceleration rate of change of velocity
a
m s⁻²
+ or −
The key pattern: every quantity that includes a direction (displacement, velocity, acceleration) can be negative. Every quantity that’s a magnitude (distance, speed) is always positive.
Distance vs displacement — the trap
These two quantities sound similar but mean different things. Imagine a particle that moves along a number line — it goes 10 m to the right, then 6 m back to the left.
• Distance travelled: 10 + 6 = 16 m (total path, ignoring direction)
• Displacement: 10 − 6 = 4 m (final position relative to start)
📍
Returning to start ≠ zero distance
If a particle returns to its starting point, its displacement is zero — but the distance travelled is not. They’re equal only when the motion never reverses.
Speed vs velocity
Speed is just the size of velocity. Drop the sign and you’re left with speed:
Speed = magnitude of velocity
If v = 7 m s⁻¹ → speed = |v| = 7 m s⁻¹
If v = −5 m s⁻¹ → speed = |v| = 5 m s⁻¹
So a particle moving “left at 5 m/s” has velocity −5 but speed 5. Both describe the same motion — velocity tells you direction, speed doesn’t.
Speeding up vs slowing down — the sign rule
“Negative acceleration” doesn’t always mean “slowing down”. What matters is the relationship between velocity and acceleration:
Comparing v and a — same or different signs?
speeding up
v · a > 0
same sign
e.g. v = 5, a = 2 (both +) or v = −3, a = −1 (both −)
slowing down
v · a < 0
opposite signs
e.g. v = 4, a = −1 (different) or v = −2, a = 3 (different)
🧠
“Same signs work together; different signs fight”
If acceleration helps the velocity (same sign), the particle speeds up. If acceleration opposes the velocity (different sign), the particle slows down. The direction of motion comes from v alone, not a.
Reading a velocity-time graph
A velocity-time graph packs four kinds of information into one picture. Knowing what each feature means lets you answer most kinematics questions without any calculation.
Anatomy of a velocity-time graph
Four key features:
• Gradient of the line at any point = the acceleration at that time.
• Where the graph crosses the t-axis (v = 0) = the particle is momentarily at rest.
• Area above the t-axis = forward movement; area below = backward movement.
• Displacement = (above) − (below); Distance = (above) + (below).
Common exam phrases
📖
Quick translation guide
“At rest” → v = 0 | “Initially” / “At the start” → t = 0 | “Returns to start” → s = 0 | “Moving right / due east” → v > 0 | “Moving down / dropped” → v < 0 | “Constant velocity” → a = 0
Worked examples
WE 1
Distance vs displacement
A particle moves along a straight horizontal line. Starting at the origin, it travels 12 m to the right, then 18 m to the left, then 4 m to the right, before coming to rest. Find:
(a) the total distance travelled, (b) the displacement from the origin at the end.
part (a) — distanceadd up all the path lengths, ignoring directiond = 12 + 18 + 4 = 34distance = 34 mpart (b) — displacementuse signs: right = +, left = −s = +12 − 18 + 4 = −2displacement = −2 m (i.e. 2 m to the left of origin)distance is always positive, displacement keeps the direction!
WE 2
Speeding up or slowing down — the sign rule
At four different times, a particle has the velocity and acceleration values shown. For each, state (i) the direction of motion, (ii) whether the particle is speeding up or slowing down.
(a) v = 6, a = 3 | (b) v = 4, a = −2 | (c) v = −5, a = −1 | (d) v = −3, a = 4
(a) v = 6, a = 3v > 0 → moving rightsame signs (both +) → speeding up(b) v = 4, a = −2v > 0 → moving rightdifferent signs → slowing down(c) v = −5, a = −1v < 0 → moving leftsame signs (both −) → speeding up(d) v = −3, a = 4v < 0 → moving leftdifferent signs → slowing downdirection comes from v · speeding/slowing comes from comparing signsnever assume “negative acceleration = slowing down” — always compare with v!
WE 3
Reading a velocity-time graph
A particle moves in a straight line. Its velocity v (m s⁻¹) over the first 8 seconds is shown:
Find: (a) the time(s) when the particle is at rest, (b) total displacement after 8 seconds, (c) total distance travelled.
part (a) — at restat rest where v = 0 (graph touches t-axis)at t = 0 and t = 6 secondspart (b) — displacement (signed area)break into shapes:0 to 2: triangle, ½ × 2 × 8 = 82 to 4: rectangle, 2 × 8 = 164 to 6: triangle, ½ × 2 × 8 = 86 to 8: triangle (below), ½ × 2 × 4 = 4displacement = 8 + 16 + 8 − 4 = 28displacement = 28 mpart (c) — distance (all positive)distance = 8 + 16 + 8 + 4 = 36distance travelled = 36 mbelow-axis area subtracts for displacement, but adds for distance!
💡 Top tips
Always state the sign convention — write down “right is positive” or “up is positive” at the start of every question.
Distance is always positive. If you compute a negative distance, you’ve made a sign error.
Speed is always positive. If a question gives speed, it’s a magnitude — assign a sign yourself based on direction.
Compare v and a to decide speeding up vs slowing down — never assume from a alone.
On a v-t graph: gradient = acceleration, area = displacement (signed) or distance (unsigned).
Sketch a v-t graph if one isn’t given — your GDC can plot it from a velocity expression.
⚠ Common mistakes
Confusing distance and displacement. If the particle reverses, they’re different.
Assuming negative acceleration = slowing down. Compare with the sign of v first.
Ignoring sign in velocity — “the particle moves at 5 m/s left” means v = −5, not +5.
Mixing units — make sure t is in seconds and distances are in metres.
Using v-t graph areas without sign awareness when computing displacement — areas below count as negative.
Confusing “at rest” with “no acceleration” — “at rest” means v = 0; “constant velocity” means a = 0.
Next up: Calculus for Kinematics — using differentiation to get velocity from displacement (and acceleration from velocity), and using integration to go the other way. The vocabulary you’ve just learned now becomes the language of equations.
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