Particles, Frames & Displacement
Welcome to Describing Motion. This topic gives you the words and tools physics uses to describe how things move: where an object is, how far it has gone, how fast it is moving, how quickly its velocity changes, and how to read all of this from a graph. It appears on every paper. Of the 192 questions on the four real papers from December 2025 to April 2026, 13 came from this topic, two to four a paper, and every one of them sat between Q6 and Q20, among the shorter questions near the start. They reward exact definitions far more than algebra.
This page sets out the definitions the rest of the topic uses. When can a whole car, or a whole planet, be treated as a single point? When we say something is moving, moving relative to what? What is the difference between "at the end of the 3rd second" and "in the 3rd second"? And why is the distance you travel not always how far you end up from where you started? Five real questions come straight from this page, January 2026 Q6 and Q7, December 2025 Q14, March 2026 Q18 and April 2026 Q20, and each is a quick mark once the definitions are clear.
Treating a body as a particle
Suppose you want to know how long a car takes to drive 300 km from one city to the next. The car is about 4 m long, but for this question that hardly matters. Compared with 300 km, the car is a dot on the map, and every part of it, the front bumper, the back seat and the roof, makes the same journey at the same time. To describe that motion, it is enough to follow one point.
Now suppose the same car is reversing into a tight parking space. This time its length and width are the whole question: whether the back bumper touches the wall depends on where each part of the car is. One point is no longer enough.
So whether a body can be replaced by a single point depends on the question you are asking, not on the body alone. That point, which carries all of the body's mass but has no size or shape, is called a particle. Size compared with the distances involved is one thing to look at; rotation is another. The Earth is enormous, yet when you study its yearly journey round the Sun, about 150 million km away, it can be treated as a particle. When you study how its spin makes day and night, it cannot: a point has no sides, so it cannot turn one side towards the Sun and then the other.
Table tennis is very suitable for the characteristics of Asians and is the national sport of China. Which of the following statements about table tennis is correct?
A. Ping Pong is relatively small and can be viewed as a particle at any time
B. When studying the flight path of the ball, it can be viewed as a particle
C. The displacement of the ball during its flight in the air is equal to the distance traveled
D. When studying the rotation of the ball, one must consider it as a particle
Test each statement by asking whether the ball's size and shape matter for the question in it.
A uses the trap above: being small is not the test. "At any time" includes the times when a player studies the ball's spin, and then its shape matters. False.
B asks only where the ball goes. The ball is 4 cm across and travels a few metres from bat to bat, so its size makes no difference to the curve it follows; following its centre is enough. True.
D asks about rotation, which means different parts of the ball moving in different directions round its centre. A point has no parts, so it cannot rotate. To study the spin, the ball must not be treated as a particle; D says the opposite. False.
C is about distance and displacement, which this page defines further down. We come back to it there, and it turns out to be false too. So the correct statement is B.
Moving relative to what? Reference frames
Sit in a train moving steadily along a straight track and look at the passenger in the seat opposite you. Is she moving? To you, no: she stays in the same seat, the same distance from you. To someone standing beside the track as the train goes by, yes: she rushes past at the speed of the train. Neither observer is wrong. They are describing her motion relative to different bodies: you relative to the train, the person outside relative to the ground.
Every description of motion works like this. To say that something moves, or stays still, you must say relative to which body, and that body is treated as being at rest. It is called the reference frame. Change the frame and the description changes. Look out of the train window and the trees beside the track slide past backward. Relative to the train, they really do move backward; relative to the ground, they stand still, and it is the train that moves forward.
To find the frame that a description uses, ask: relative to which body does the object move in the way described? One answer can always be ruled out. A body is always at rest relative to itself, so the object being described can never be the frame in which it moves.
When sitting in a train moving at a constant speed, the trees outside the window appear to move backward. The reference frame chosen is ( )
A. The Sun B. The trees C. The ground D. The train
Ask what the trees move backward relative to. You are sitting in the train, and the trees slide backward past its windows: relative to the train, they move backward. So the description uses the train as its reference frame, and the answer is D.
A: relative to the Sun, the trees are carried round by the Earth's spin and its orbit. That motion has nothing to do with the train, so it is not "backward".
B: the trees cannot be the frame in which the trees move, because a body is always at rest relative to itself.
C is the trap. The ground is the usual frame, but relative to the ground the trees stand still and the train moves forward. That describes the train's motion, not trees moving backward.
Instants and time intervals
The word "time" is used for two different things. "The train leaves at 8:00" names a single moment, an instant. "The journey takes 2 hours" names a stretch of time between two instants, a time interval. Draw time as an axis, like a number line, and the difference is easy to see: an instant is a point on the axis, and a time interval is a segment between two points, with a length.
Physics questions start the clock when the motion starts, at t = 0, and count seconds from there. The 1st second is the interval from t = 0 to t = 1 s; the 2nd second runs from 1 s to 2 s; in general, the nth second runs from t = (n − 1) s to t = n s, and every one of them lasts 1 s. "The first 3 s", or "in 3 s", is the interval from 0 to 3 s, which lasts 3 s. "At the end of the 3rd second" is not an interval at all: it is the instant t = 3 s, the same instant as "the start of the 4th second".
| The wording | What it names |
|---|---|
| "at the end of 3 s", "at the end of the 3rd second" | The instant t = 3 s |
| "at the start of the 4th second" | The same instant, t = 3 s |
| "in the 3rd second", "during the 3rd second" | The interval from t = 2 s to t = 3 s, lasting 1 s |
| "in the first 3 s", "in 3 s" | The interval from t = 0 to t = 3 s, lasting 3 s |
A ball starts rolling along a straight track at t = 0. Say whether each phrase names an instant or an interval, and place it on the time axis: (1) the end of the 2nd second; (2) the 2nd second; (3) the first 2 s; (4) the start of the 3rd second.
(1) An instant. The 2nd second finishes at t = 2 s.
(2) An interval. The 2nd second runs from t = 1 s to t = 2 s, and lasts 1 s.
(3) An interval, from t = 0 to t = 2 s, lasting 2 s. It contains both the 1st second and the 2nd second.
(4) An instant. The 3rd second begins at t = 2 s, so (1) and (4) are the same instant, named in two different ways.
Distance and displacement
Walk from home 400 m east to a shop, then 300 m north to a park. How far have you gone? That question has two honest answers. Your feet covered 400 m + 300 m = 700 m: that is the length of your path. But the park is not 700 m from your home. In a straight line it is √(400² + 300²) = 500 m away, in a direction between north and east. If you want to know how much walking you did, 700 m is the answer; if you want to know where you ended up compared with where you started, you need the 500 m and its direction.
Physics gives the two answers different names. The length of the path is the distance travelled. The straight arrow from the starting position to the final position is the displacement. They answer different questions, so they behave differently. Take the long way round to the park and the distance grows, but the displacement does not change: it depends only on where you started and where you finished. And because the displacement points somewhere, it has a direction as well as a size, while the distance does not.
For motion along a straight line, put the positions on a number line, the x-axis. The displacement is then Δx = x₂ − x₁, the final position minus the starting position, and its sign gives its direction. Take upwards as positive: a lift starts at the ground floor, x = 0, rises to x = 12 m, then comes back down to x = 6 m. Its displacement is Δx = 6 m − 0 = +6 m, so 6 m upwards, but the distance it travelled is 12 m + 6 m = 18 m.
Can the distance and the size of the displacement ever be equal? The straight line is the shortest route between two points, so no path can be shorter than the displacement arrow: the size of the displacement is never more than the distance. The two are equal only when the path is that straight line itself, which means motion along a straight line in one direction, never turning back. Any bend in the path, or any turning back, makes the distance larger.
Which of the following statements about distance and displacement is correct?
A. The magnitude of displacement and the distance traveled can be equal
B. The magnitude of displacement and the distance traveled are never equal
C. Distance is a vector; displacement is a scalar
D. Both distance and displacement are vectors
A says "can be", so one example is enough. A car that drives 2 km along a straight road without turning back has a displacement of 2 km and a distance of 2 km. True.
B is the opposite of A, and the same car proves it false. It is the statement you would choose if you remembered only that the two are usually different.
C swaps the two. Displacement, the arrow from start to finish, has a direction and is the vector; distance, a length of path, is the scalar.
D gives distance a direction it does not have. A path length of 700 m is complete without saying which way.
So the correct statement is A.
Now we can finish January 2026, Q6. Option C said that the ball's displacement during its flight equals the distance it travels. A ball crossing a table-tennis table rises and then falls, so its path is a curved arc, longer than the straight arrow from where it was hit to where it lands. Its distance is greater than the size of its displacement, so C is false, and B is the only correct statement.
Round a circle
Circular paths are the exam's favourite way of pulling distance and displacement apart. Take a body moving round a circle of radius R, starting at a point P, and follow it.
After a quarter turn, its path is a quarter of the circumference: s = ¼ × 2πR = πR/2. Its displacement is the straight chord from P to where it stops. The radii to these two points are at right angles, so Pythagoras gives the chord as √(R² + R²) = √2 R.
After half a turn, it reaches the point directly opposite P. The path is half the circumference, s = πR. The displacement arrow runs straight across through the centre: it is a diameter, so its size is 2R.
After one full turn, it is back at P. The path is the whole circumference, s = 2πR. But the body has finished where it started, so the displacement arrow has no length at all: the displacement is 0. Here the gap between the two quantities is as large as it gets: a long path, and no displacement.
The radius of a Ferris wheel is R. The cabin rotates with the wheel once, and its distance and displacement are
(The paper shows a Ferris wheel turning, with one cabin marked on its rim.)
A. distance 2πR, displacement 2R B. distance 2R, displacement 2πR
C. distance 2R, displacement 0 D. distance 2πR, displacement 0
A particle moving along a circular path of radius r completes one revolution and returns to the original position. Its displacement and distance traveled are, respectively, ( ).
A. 0, 0 B. 2πr, 0 C. 0, 2πr D. 2πr, 2πr
Both questions describe one full turn of a circle, so the physics is the same. The path is the whole circumference, 2πR, and the body ends where it began, so the displacement is 0. (The cabin's size does not matter for this question, so it can be treated as a particle on the rim.)
January asks for the distance first, then the displacement: 2πR and 0, which is D.
March asks in the other order, "displacement and distance, respectively": 0 and 2πr, which is C.
The wrong options are built from the usual slips. In January, A and C both use 2R, the size of the displacement after half a turn, not a full one, and B gives the displacement the length of the path. In March, B holds exactly the right two values in the wrong order; A forgets that a body that has gone round a circle has travelled some distance; and D gives the displacement the length of the path.
Vectors and scalars
Displacement is the first quantity in this topic that needs a direction to be complete. "The park is 500 m from home" leaves a question open: which way? "500 m, in a direction between north and east" answers it. A time, a mass or a distance leaves no such question: 5 s, 2 kg and 700 m are complete as a number with a unit.
Quantities that need a direction are vectors; quantities that are complete without one are scalars. You have already met one vector, force, in Weight, Springs & Friction. On the next page you will meet velocity, which says how fast something moves and in which direction: it is a vector, while speed, which says only how fast, is a scalar. Later in this topic comes acceleration, which is also a vector.
Vectors also combine in their own way. The two legs of the walk, 400 m east and 300 m north, gave a displacement of 500 m, not 700 m: the arrows join tip to tail, and the result is the arrow from the first tail to the last tip. Distances, being scalars, simply add as numbers.
Statement questions: putting it together
Several exam questions on this page's ideas are statement questions: four statements, and you choose the correct one, or the incorrect one. Test each statement against four checks.
Before you test anything, read the stem again. Does it ask for the correct statement or the incorrect one? The CSCA uses both.
Your turn! · Practice questions
Things to remember
| Concept | What to know |
|---|---|
| Particle model | A point with the body's mass but no size or shape. Use it when size and shape do not affect the question: a ball's flight path, yes; its spin, no. Being small is not the test. |
| Reference frame | The body that motion is described relative to, treated as at rest. The same motion looks different in different frames; when no frame is named, use the ground. |
| Finding the frame | Ask what the object moves relative to. Trees moving backward, seen from a train, use the train as the frame. A body is always at rest relative to itself. |
| Instant and interval | An instant is a point on the time axis: the end of the 3rd second = the start of the 4th = t = 3 s. An interval is a segment: the 3rd second is 2 s to 3 s; the first 3 s is 0 to 3 s. |
| Distance | The length of the path travelled. A scalar, measured in metres. |
| Displacement | The straight arrow from start to finish. A vector, measured in metres, that depends only on the start and the end. On a line, Δx = x₂ − x₁, and its sign gives the direction. |
| Comparing them | |Δx| ≤ s, equal only for straight-line motion that never turns back. Any bend or turning back makes the distance larger. |
| Round a circle of radius R | Distance, then displacement: quarter turn πR/2 and √2 R; half turn πR and 2R; full turn 2πR and 0. |
| Answer order | Check whether the stem asks for "distance and displacement" or "displacement and distance"; the right pair in the wrong order is often an option. |
| Vectors and scalars | Vectors: displacement, velocity, acceleration, force. Scalars: distance, speed, time, mass. |