Module 3: Kinematics · Theory · Prerequisites: K3, Foundation 8
An average describes an interval. To describe an instant, the interval has to disappear.
Mahavakya
A moving object does not carry a number called velocity inside it. Velocity is something we construct from how its position changes.
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Learning objectives
After this reading you should be able to:
- Obtain instantaneous velocity as the limit of an average, and read it as the slope of a tangent.
- Differentiate a position function to get velocity and acceleration.
- Decide whether a body is speeding up or slowing down from the signs of v and a, using one criterion rather than four cases.
- List what v = 0 can mean, and test whether a particular instant is a turning point.
- Say why a = 0 does not imply v = 0, and why the reverse fails too.
1. Shrinking the interval
K3 gave the cyclist an average of 3.33 m/s over six minutes. That number says nothing about how fast she was going after ninety seconds, and no amount of care with the average will recover it.
The way out is to make the interval smaller. Take a position function x(t) = t², and ask for the average velocity over intervals that all begin at t = 1 but end progressively sooner:
1 to 4: (16 − 1)/3 = 5 1 to 3: (9 − 1)/2 = 4 1 to 2: (4 − 1)/1 = 3
1 to 1.1: (1.21 − 1)/0.1 = 2.1 1 to 1.01: 2.01
The numbers are settling on 2. Not reaching it — every one of them is an average over a genuine stretch of time, and none is a velocity at an instant. But they are approaching something, and that something is what we call the instantaneous velocity:
v(t) = limΔt→0 Δx/Δt = dx/dt
This is Foundation 8’s limit, met again in a physical setting. What was there a construction — the chord becoming a tangent — is here the answer to a question about a moving body.
2. Secant and tangent
On a position–time graph the two averages have shapes.
Average velocity over shorter and shorter intervals: 5, 4, 3 … approaching 2.
Average velocity is the slope of a secant — the straight line joining two points on the curve. Instantaneous velocity is the slope of the tangent — the line touching the curve at one point.
Shrinking the interval swings the secant toward the tangent, and in the limit they coincide. That is the whole of the definition, drawn.
Worked example
A particle moves with x(t) = 4t² + 2t + 1 metres. Find its velocity at t = 2 s, and its average velocity over the first two seconds.
Differentiating: v(t) = 8t + 2, so v(2) = 18 m/s.
For the average, x(0) = 1 and x(2) = 16 + 4 + 1 = 21, so
vavg = (21 − 1)/2 = 10 m/s
Ten against eighteen. The average is smaller because the particle was slower for most of the interval — it began at 2 m/s and finished at 18. Neither number is wrong; they describe an interval and an instant.
3. When the two averages agree
Sometimes the average velocity over an interval equals the instantaneous velocity at its midpoint. It is worth seeing this happen, and then seeing it fail.
Take x = t² + 2t over the interval from 1 to 3. The average is (15 − 3)/2 = 6 m/s. The instantaneous velocity is v = 2t + 2, so at the midpoint t = 2 it is 6 m/s. They agree exactly.
Now take x = t³ over the same interval. The average is (27 − 1)/2 = 13 m/s. The instantaneous velocity is v = 3t², so at t = 2 it is 12 m/s. They do not agree.
The first case was a coincidence, not a rule. It works when the velocity itself changes at a constant rate, so the speeding-up before the midpoint exactly compensates the speeding-up after. For t² the velocity is linear and it does; for t³ the velocity is quadratic and it does not.
A student who meets only the first example concludes that average and instantaneous velocity are interchangeable at midpoints. They are not, and the second example is the reason to see both.
4. Instantaneous acceleration
Everything just done to position can be done again to velocity:
a(t) = dv/dt = d²x/dt²
Acceleration is the rate of change of velocity, and velocity is a vector — so acceleration responds to a change in direction as much as to a change in speed. In one dimension a direction change appears as a change of sign, which keeps things manageable. In two dimensions it will not, and a body can accelerate while its speed never varies. That is K15’s business.
Two consequences worth stating now, because both are routinely got wrong.
a = 0 does not mean v = 0. Zero acceleration means the velocity is not changing — which is exactly what happens when a body moves steadily at 30 m/s. Constant velocity and zero velocity both give a = 0, and they are different situations.
v = 0 does not mean a = 0. A ball at the top of its flight has zero velocity and an acceleration of g downward. If the acceleration were zero it would hang there.
5. Speeding up and slowing down
Here is a claim students believe for years: negative acceleration means slowing down.
It is true when the velocity is positive, and false otherwise. A ball falling downward, with upward taken as positive, has a = −g and is getting steadily faster.
The correct criterion is one line. Speeding up means the speed is increasing, that is d|v|/dt > 0. In one dimension that condition is exactly:
va > 0 → speeding up
va < 0 → slowing down
Same sign, speeding up. Opposite signs, slowing down. That is the whole rule, and it is worth having in this form rather than as four remembered cases — because the four cases are what it produces:
| v | a | Motion | Example |
|---|---|---|---|
| + | + | speeding up | car accelerating forward |
| + | − | slowing down | ball rising |
| − | + | slowing down | car braking while reversing |
| − | − | speeding up | ball falling |
Worked example
A particle has v(t) = t² − 6t + 5 m/s. Describe its motion.
Factorise: v = (t − 1)(t − 5), so v = 0 at t = 1 and 5. And a = 2t − 6, so a = 0 at t = 3. Three critical times, four intervals:
| 0 < t < 1 | v > 0, a < 0 | slowing |
| 1 < t < 3 | v < 0, a < 0 | speeding up |
| 3 < t < 5 | v < 0, a > 0 | slowing |
| t > 5 | v > 0, a > 0 | speeding up |
The pattern alternates, and it is not guessable — you have to hold both signs at once across four regions. Note also that at t = 3 the acceleration vanishes while the particle is moving at its fastest in the negative direction. Zero acceleration at maximum speed, which is exactly what you would expect if you think about what acceleration is.
6. Four things v = 0 can mean
Finding the times at which the velocity vanishes is usually easy. Interpreting them is where marks are lost, because v = 0 admits four quite different readings:
- A turning point. The velocity changes sign, and the body reverses.
- A momentary pause. The velocity touches zero and returns to the same sign. The body slows to a stop and carries on the way it was going.
- A start from rest. The body was not moving and begins to.
- Permanent rest. The body has stopped and stays stopped.
Only the first is a turning point, and nothing about the equation v = 0 distinguishes them. The distinction is in what the velocity does either side.
7. Turning points, properly
So the rule is this:
v = 0 is necessary for a turning point, but not sufficient. The velocity must also change sign there.
The cleanest counterexample is x = t³, for which v = 3t². At t = 0 the velocity is zero. Just before, v = 3t² > 0. Just after, v = 3t² > 0 again. The particle slows to an instantaneous stop and continues forward without ever reversing.
Squared factors are the warning sign. If v factorises with a repeated root — something of the form (t − c)² — the velocity touches zero at t = c without crossing, and there is no turning point there however much the algebra looks like there should be.
The test is one line. Having found v = 0 at some time, evaluate v a little before and a little after. If the signs differ, it is a turning point. If they match, it is not. Do not reason from the sign of the acceleration — that will mislead you at a repeated root.
Contemplation
There is something strange in the phrase “velocity at an instant”. An instant has no duration. Nothing moves during it. Take a photograph of a moving car and the car in the photograph is not moving — and yet we say the car has a velocity at the moment the photograph was taken.
The resolution is that the object does not carry a number called velocity inside it. Velocity is not a property we find; it is a description we construct, out of how the position behaves in the neighbourhood of that instant. The limit is what does the constructing. The instant supplies a location and nothing else.
Zeno noticed the difficulty and had no way to answer it. Foundation 8 gave the answer, and it is worth appreciating what it cost: two thousand years, and a piece of mathematics built for the purpose.
Common misconceptions
1. Negative acceleration means slowing down
Only when the velocity is positive. Use va > 0 for speeding up; the sign of a alone decides nothing.
2. v = 0 means a turning point
It is necessary and not sufficient. Check the sign either side. A repeated root gives zero velocity with no reversal.
3. a = 0 means the body is at rest
It means the velocity is not changing. A body cruising at 30 m/s has zero acceleration and is not remotely at rest.
4. Position is large, so velocity is large
Position is the height of an x–t graph; velocity is its slope. A body a hundred metres from the origin may be perfectly still.
5. Average velocity equals instantaneous velocity at the midpoint
True only when velocity changes at a constant rate. Section 3 shows it holding for t² and failing for t³.
Reflection
- In the four-interval example, the acceleration vanishes at t = 3 while the particle moves fastest. Why should that not be surprising?
- Can a body have zero velocity and zero acceleration at the same instant and still be about to move? What would have to be true?
- The tangent is the limit of secants. What, physically, is the secant a limit of?
Key takeaways
- Instantaneous velocity is the limit of average velocity as the interval shrinks: v = dx/dt.
- Average velocity is the slope of a secant; instantaneous velocity the slope of a tangent.
- Acceleration is dv/dt, and responds to changes in direction as well as speed.
- va > 0 means speeding up; va < 0 means slowing down. One criterion, four cases.
- a = 0 does not mean v = 0, and v = 0 does not mean a = 0.
- v = 0 is necessary but not sufficient for a turning point. Check the sign either side.
What comes next
Every quantity in this part was extracted from a formula by differentiating. But most of what you have just learned is visible on a graph without any calculation at all — the slope is the velocity, and its steepening or flattening is the acceleration. K5 reads motion straight off the position–time graph.
Prerequisites: K3 and Foundation 8 · Reading: 16 min · Practice: 30 min · Difficulty: Intermediate