Intuition The one core ideaAcceleration is how fast the velocity arrow is changing — and we can ask that question two ways: over a stretch of time (average) or at one frozen instant (instantaneous). Everything on the parent page is just these two questions, dressed up in the symbols Δ lim d / d t
Before you can read the parent note honestly, you must own every mark it writes. Below, each symbol is built from nothing: plain words → a picture → why the topic can't live without it. They are ordered so each one leans on the one before.
Position is where a thing is . In one dimension we pin it to a number line: pick a zero (the origin), pick a positive direction, and the object's location is a single number we call x
Look at figure 1. The amber dot sits at some mark on the ruler. That mark is x x which side .
Intuition Why a symbol at all?We write x changes with time , and we want to feed it into machinery (subtraction, slopes, limits). A word can't be subtracted; a number can.
When motion is 2-D or 3-D we bundle the coordinates into one vector and write r Vectors: addition and subtraction .
t Δ t t is the reading on a clock, in seconds (s). The Greek capital delta Δ "the change in" — literally final value minus initial value .
Δ t = t f − t i Δ x = x f − x i i f
Δ Δ length of a gap . In figure 1, Δ x Δ t Δ always a subtraction — end minus start, never the other way.
Δ vector (an arrow), Δ final minus initial , but the minus is now a vector subtraction :
Δ r = r f − r i , Δ v = v f − v i r i Vectors: addition and subtraction ; figure 4 shows it in action for velocities.
Δ Δ Why it feels right: Δ t 2 t The fix: Δ Δ t single quantity meaning "the amount the clock advanced." You cannot cancel the Δ Δ
Velocity is how fast position changes, and in which direction . Over a time interval it is the vector change in position divided by the elapsed time:
v a v g = Δ t Δ r = t f − t i r f − r i v = d r / d t
Look at figure 2, a position–time graph : time runs along the bottom, position up the side, and the object's history is a curve. The velocity between two moments is the steepness (slope) of the straight line joining those two points: steep = moving fast; flat = standing still; sloping down = moving in the negative direction. This is exactly the Average velocity vs instantaneous velocity story — the parent topic sits one rung above it.
Why do we need velocity before acceleration? Because acceleration is the change of velocity . You cannot ask "how fast is the velocity changing?" until you can measure the velocity itself. Position → velocity → acceleration is a ladder; you climb it one rung at a time.
Slope = (how much the vertical thing changed) ÷ (how much the horizontal thing changed) = "rise over run." For a straight line between two points it is a single number.
Definition Secant and tangentA secant is a straight line that cuts a curve at two points — it summarises the change across the gap between them. A tangent is a straight line that just touches the curve at one point , matching its direction there — it captures the behaviour at that single instant .
For the rest of this section we restrict to one dimension (motion along a single line), so the velocity vector v v
Look at figure 3. On a velocity–time graph:
The cyan chord joining two points on the curve is a secant . Its slope Δ t Δ v average acceleration (1-D).
The amber line just kissing the curve at one point is the tangent . Its slope = instantaneous acceleration (1-D).
Intuition Why two kinds of line?A secant needs two points, so it answers "what happened across the gap." A tangent lives at one point, so it answers "what's happening right here, right now." That single distinction is the entire parent topic in geometric form.
==Δ t → 0 lim watch what a quantity approaches as Δ t — without ever plugging in exactly zero (which would give the forbidden 0/0
Intuition Picture of the limitIn figure 4, keep the left point fixed and slide the right point toward it. The secant swings, and as the gap Δ t secant rotates until it lies flat against the tangent . The limit is the destination of that swinging line. "Average over a shrinking window" morphs into "value at the instant."
Δ t = 0 Why it feels right: if we want the value at a point, why not use Δ t = 0 The fix: Δ t = 0 Δ t Δ v = 0 0 approaching zero and reading off the trend, never landing on it.
Once the limit above settles down to a single number for every instant, we give it a name and a compact symbol — the derivative :
a = d t d v = lim Δ t → 0 Δ t Δ v d t d v v d not fractions you cancel; the whole symbol is one idea: instantaneous rate of change . See Derivatives as rates of change . This same symbol also lets us finally write the instant-by-instant velocity promised in Section 3, v = d r / d t
Units: velocity is in m/s and we divide by time in s, so acceleration is measured in metres per second per second , written m/s 2
Intuition The sign is a direction, not "speeding up or slowing down"Once you pick a positive direction on your line, acceleration inherits a sign. But that sign tells you which way Δ v not whether the object is speeding up. You must compare it to the sign of the velocity:
a v same sign → the object is speeding up (e.g. v > 0 a > 0 a v opposite signs → the object is slowing down (e.g. v > 0 a < 0 a = 0
Worked example Same magnitude, opposite meaningA car moving in the + a = − 3 m/s 2 braking . A car moving in the − same a = − 3 m/s 2 speeding up backwards . Same number, opposite physical story — because the velocity's sign differs. Always read a relative to v
Intuition The arrow is not optionalv Δ v = v f − v i vector subtraction (Section 2) — you may have to flip and add arrows, not just subtract numbers (see Vectors: addition and subtraction ). This is why a car going round a curve at constant speed still accelerates: the speed (arrow length) is fixed, but the arrow keeps turning, so Δ v = 0 Uniform circular motion .
Figure 5 shows this: two velocity arrows of equal length but different direction, and the amber Δ v
Number line and position x
Time t and the change symbol delta
Vectors have size and direction
Velocity = change of position per time
Limit as delta t goes to zero
Derivative = instantaneous rate
Instantaneous acceleration
What does the symbol Δ "The change in" — always final minus initial (Δ t = t f − t i
How does Δ r Δ r = r f − r i — a vector subtraction (subtract the arrows, not just lengths).
On a position–time graph, what does slope represent? The velocity (steepness = how fast position changes).
What is a secant line? A straight line cutting a curve at two points; its slope gives the average rate across the gap.
What is a tangent line? A straight line touching a curve at one point; its slope gives the instantaneous rate there.
On a velocity–time graph, what does the slope of a secant give? Average acceleration Δ v /Δ t
On a velocity–time graph, what does the slope of a tangent give? Instantaneous acceleration.
What are the units of acceleration? Metres per second squared, m/s 2
What does lim Δ t → 0 Watch what the quantity approaches as the gap shrinks toward zero, without setting it exactly to zero.
Why can't you just set Δ t = 0 Δ v /Δ t You would get 0/0
What is a derivative in plain words? The instantaneous rate at which one quantity changes per unit of another.
When is an object with negative acceleration actually speeding up? When its velocity is also negative — same sign as a
Why is a It is a vector; velocity has direction, so direction changes also count as acceleration.
What does the "2" mean in d 2 r / d t 2 Differentiate position twice with respect to time (a counter, not a power).
Why must you learn velocity before acceleration? Acceleration is the rate of change of velocity — the ladder position → velocity → acceleration.