Module 3: Kinematics · Orientation · Prerequisite: Foundations 2 and 3
Before anything can be said about how a body moves, someone has to choose where to watch it from.
Mahavakya
Motion is not a property of a body. It is a relation between a body and a frame someone chose.
On this page
Learning objectives
After this reading you should be able to:
- Say why “the body is at rest” is an incomplete statement.
- State what a reference frame consists of, including the part most students omit.
- Distinguish moving the origin from changing the frame, and say what each leaves unchanged.
- Compute distance and displacement for a journey with a reversal in it.
- Prove, rather than assert, that distance can never be less than the magnitude of displacement.
- Use signs in one dimension without confusing direction with size.
1. There is no absolute rest
A passenger sits still in a moving train.
Relative to the train, the passenger is at rest. Relative to the platform, the passenger moves at the speed of the train. Relative to the centre of the Earth, the passenger is also being carried around by the planet’s rotation. Relative to the Sun, the passenger travels along a curve at some thirty kilometres per second.
One person, one moment, four descriptions. None of them is the true one, and none is wrong.
So the sentence “the passenger is at rest” is not false. It is incomplete. It becomes a physical statement only when it says at rest relative to what.
Nothing in mechanics allows you to identify absolute rest. There is no experiment that finds the frame in which the universe is truly still, and physics stopped looking a long time ago. Every statement about motion carries a frame, whether or not it is mentioned.
2. What a reference frame is
A reference frame is four things:
- an origin, a point chosen as zero;
- axes, giving directions to measure along;
- a clock, so that events can be given a time;
- a rule for measuring, so that positions and times are numbers rather than impressions.
The clock is the one students leave out, and it is worth keeping. A frame is not merely a set of axes drawn on paper; it is a way of assigning a position and a time to every event. In Newtonian mechanics the clock is the same for everyone, so it can be forgotten without immediate harm. That will not always be true, and the habit of including it costs nothing now.
Notice also what a frame is not. It is not part of the physical world. Foundation 3 said this about coordinate systems, and it holds here with more force, because a frame can be attached to a moving object. The train has no axes painted on it. We chose to describe the passenger from the train, and the description followed from the choice.
3. Position needs an origin
Once a frame is fixed, the position of a particle is the vector from the origin to the particle:
r = xî + yĵ + zk̂
In one dimension this reduces to a single number x, and the sign carries the direction.
Two things are worth separating here, because they are routinely confused.
The coordinates specify the position; the position vector represents it. And x may be negative — a particle three metres to the left of the origin has x = −3 m — while the distance from the origin, |x| = 3 m, cannot be. One is a location with a direction built in. The other is a length.
4. Moving the origin, and changing the frame
These are two different operations, and telling them apart now will save a great deal of confusion when relative motion arrives.
Moving the origin means keeping the same frame and relabelling. Shift the origin three metres to the right and every position changes by three metres. Nothing physical has happened at all.
The same two points, described from two origins.
In symbols, if the origin moves by d:
xnew = x − d
but vnew = v and anew = a
Positions change. Velocity and acceleration do not, because d is a constant and differentiating removes it.
Changing the frame is a different matter. Watch the same particle from a frame moving at constant velocity u, and the velocity you measure changes:
vnew = v − u
but still anew = a
Now the velocity does change, and the acceleration still does not — because u is constant, and differentiating a second time removes it too.
Acceleration is the most stable quantity here. Position depends on where you put the origin. Velocity depends on how your frame is moving. Acceleration survives both. That is not a coincidence, and it is the reason Newton’s second law is written in terms of acceleration rather than velocity.
5. Distance and displacement
Displacement is the change in position:
Δr = rfinal − rinitial
It is a vector, it depends only on the endpoints, and it is unaffected by where the origin sits — as the figure above shows, since the origin cancels in the subtraction.
Distance, written s, is the total length of the path actually travelled. It is a scalar, it can never decrease as the journey continues, and it depends on the whole route.
Worked example
A particle starts at the origin, moves to x = +8 m, then back to x = +3 m, then forward to x = +10 m. Find the distance travelled and the displacement.
Distance — add the legs, ignoring direction:
s = |8 − 0| + |3 − 8| + |10 − 3| = 8 + 5 + 7 = 20 m
Displacement — endpoints only:
Δx = 10 − 0 = +10 m
Twice as far travelled as progress made. The intermediate positions +8 and +3 appear in the distance and vanish from the displacement — which is exactly the difference between the two quantities. One remembers the journey; the other remembers only where you started and where you stopped.
6. Why the distance is never smaller
Every book states that s ≥ |Δr|. Rather less often does one say why, and the reason is worth three lines because it comes straight from Foundation 4.
Break the path into many small steps, with displacements Δr1, Δr2, …, Δrn. Then:
total displacement = Δr1 + Δr2 + … + Δrn
total distance = |Δr1| + |Δr2| + … + |Δrn|
The first is a vector sum; the second adds magnitudes. And the triangle inequality from Foundation 4 says that for any two vectors,
|A + B| ≤ |A| + |B|
Apply that repeatedly across all the steps and you get
|Δr| ≤ s
with equality only when every step points the same way — that is, motion in a straight line without ever reversing. This is not a rule about journeys. It is Foundation 4’s inequality, applied to a path.
7. Signs carry direction
In one dimension a vector has only two possible directions, so a sign is enough to record it. Choose a positive direction and the convention follows:
- x > 0 — the particle is on the positive side of the origin.
- v > 0 — it is moving in the positive direction.
- a > 0 — its velocity is increasing algebraically.
The third one is where students go wrong, so read it again. a > 0 does not mean speeding up. It means the velocity is becoming more positive — which, if the particle is moving in the negative direction, means it is slowing down.
The ball thrown upward
Take upward as positive. A ball thrown up has v > 0 and a = −g < 0. Both are correct, and the ball is going up. Negative acceleration here does not mean downward motion; it means the upward velocity is decreasing. The signs describe two different quantities and there is no rule that they must agree.
One more point about the sign convention itself: it is a choice, not a fact. Take downward as positive instead and the ball rising has v < 0 with a = +g. Every number flips and the physics is untouched. What matters is choosing once, at the start, and holding it.
Contemplation
Two particles start at the same point and finish at the same point. One travels straight there. The other wanders for an hour, doubles back twice, and arrives at the same place.
Their displacements are identical. Their distances are not. Neither number is the real answer, because they answer different questions — one asks what changed, the other asks what happened. Displacement remembers only the endpoints. Distance remembers the journey.
This pattern will return. Some physical quantities depend on the whole history and some only on the endpoints, and a great deal of mechanics turns on knowing which kind you are being asked for. Work and energy will make the same distinction later, and for the same reason.
Common misconceptions
1. “The body is at rest” is a complete statement
It is at rest relative to something. Say what. The passenger, the train, the platform and the Sun disagree, and all four are right.
2. Negative position means the particle is behind something
It means the particle is on the side of the origin you called negative. Move the origin and the sign may change without the particle moving at all.
3. Negative acceleration means slowing down
Only when the velocity is positive. A ball falling downward with upward taken as positive has a < 0 and is speeding up. The sign of the acceleration alone tells you nothing about speed.
4. Displacement can be negative, so it is not a vector
It is a vector, and in one dimension the sign is the direction. The vector itself is not positive or negative; only its component along the chosen axis is.
5. Distance equals the magnitude of displacement
Only for straight-line motion with no reversal. Any turn makes the distance strictly greater, and the worked example above gives 20 m against 10 m.
Reflection
- A frame includes a clock. Why does that not seem to matter in any problem you have met so far?
- Acceleration survives both moving the origin and changing to a frame in steady motion. What would it take to change it?
- You are told a particle’s distance travelled is 12 m. What can you say about its displacement, and what can you not?
Key takeaways
- There is no absolute rest. Every statement about motion carries a frame, stated or not.
- A reference frame is an origin, axes, a clock and a rule for measuring.
- Moving the origin changes positions and leaves velocity and acceleration alone.
- Changing to a frame in steady motion changes velocities and leaves acceleration alone.
- Displacement depends only on the endpoints; distance depends on the whole path.
- s ≥ |Δr|, from the triangle inequality, with equality only for straight unreversed motion.
- In one dimension the sign is the direction, and the sign of a alone says nothing about speeding up.
What comes next
Every idea on this page — the origin, the axes, the positive direction — is a decision you will make again in every problem from here on. And in mechanics those decisions are made on paper, in a drawing, before any equation is written.
That drawing is a physical claim, not an illustration. An arrow pointing the wrong way is as wrong as a formula with a missing sign, and rather harder to notice. The next part is about how to make one.
Prerequisite: Foundations 2 and 3 · Reading: 14 min · Practice: 20 min · Difficulty: Beginner