Walk a Full Lap and You Have Gone Nowhere
Learn to sort quantities into scalars and vectors and draw a vector properly, tell distance from displacement, calculate average speed and average velocity, and handle acceleration and retardation with the right signs.
Why is your displacement zero after running a full lap?
Run one complete lap of a m track and stop where you started. How far did you travel?
Four hundred metres — that is the distance, the total length of the path.
But your displacement is zero. Displacement asks a different question: how far are you from where you began, and in which direction? You finished at the starting line, so the answer is nothing at all.
Both numbers are correct. They answer different questions, and each is useful for something different. A runner's stamina is measured by the m; a delivery courier's usefulness is measured by the displacement.
The difference is that displacement carries a direction and distance does not. That single feature splits every physical quantity into two families — scalars, which need only a size, and vectors, which need a size and a direction. Speed is a scalar; velocity is the vector version of it. And once directions are involved, signs start to matter, which is what makes acceleration and retardation worth separating carefully.
This page covers the first part of the ICSE Class 9 Physics chapter on motion in one dimension — scalars and vectors, distance and displacement, average speed and average velocity, and acceleration.
Four hundred metres — that is the distance, the total length of the path.
But your displacement is zero. Displacement asks a different question: how far are you from where you began, and in which direction? You finished at the starting line, so the answer is nothing at all.
Both numbers are correct. They answer different questions, and each is useful for something different. A runner's stamina is measured by the m; a delivery courier's usefulness is measured by the displacement.
The difference is that displacement carries a direction and distance does not. That single feature splits every physical quantity into two families — scalars, which need only a size, and vectors, which need a size and a direction. Speed is a scalar; velocity is the vector version of it. And once directions are involved, signs start to matter, which is what makes acceleration and retardation worth separating carefully.
This page covers the first part of the ICSE Class 9 Physics chapter on motion in one dimension — scalars and vectors, distance and displacement, average speed and average velocity, and acceleration.
How do you tell a scalar from a vector, and how do you draw one?
A scalar needs only a magnitude with its unit; a vector needs a magnitude and a direction. Two quantities with the same size but opposite directions are the same scalar and different vectors.
Scalars: mass, distance, speed, time, volume, density, area, work, energy, power, temperature, pressure, electric current.
Vectors: displacement, velocity, acceleration, force, weight, momentum.
Worked classification.
- kg — scalar (mass). There is no such thing as five kilograms northward
- m — scalar if it is a distance, vector if it is a displacement. The words matter
- — scalar (speed)
- due east — vector (velocity)
- N downward — vector (weight)
- J — scalar (energy). Work and energy have no direction even though force does
How to represent a vector. Draw a directed line segment: a straight line whose length is proportional to the magnitude on a stated scale, with an arrowhead showing the direction.
Worked example. Represent a force of N acting towards the east, using the scale cm N.
So draw a cm line pointing east with an arrowhead at the eastern end. On the same scale, a force of N would be a cm arrow.
Three things a vector diagram must carry: the scale written beside it, the correct length, and the arrowhead. A line without an arrowhead is not a vector, and a diagram without its scale cannot be read.
Work and energy are scalars although force is a vector. This catches people out: force has a direction, distance has none, and their product turns out to need no direction either. So being built from a vector does not make a quantity a vector — the test is whether a direction is part of the answer, and for an energy it is not.
Speed and velocity have the same unit but are not the same quantity. A car going round a bend at a steady has constant speed and changing velocity, because its direction keeps changing. That distinction becomes the whole point of the acceleration section later on this page.
Scalars: mass, distance, speed, time, volume, density, area, work, energy, power, temperature, pressure, electric current.
Vectors: displacement, velocity, acceleration, force, weight, momentum.
Worked classification.
- kg — scalar (mass). There is no such thing as five kilograms northward
- m — scalar if it is a distance, vector if it is a displacement. The words matter
- — scalar (speed)
- due east — vector (velocity)
- N downward — vector (weight)
- J — scalar (energy). Work and energy have no direction even though force does
How to represent a vector. Draw a directed line segment: a straight line whose length is proportional to the magnitude on a stated scale, with an arrowhead showing the direction.
Worked example. Represent a force of N acting towards the east, using the scale cm N.
So draw a cm line pointing east with an arrowhead at the eastern end. On the same scale, a force of N would be a cm arrow.
Three things a vector diagram must carry: the scale written beside it, the correct length, and the arrowhead. A line without an arrowhead is not a vector, and a diagram without its scale cannot be read.
Work and energy are scalars although force is a vector. This catches people out: force has a direction, distance has none, and their product turns out to need no direction either. So being built from a vector does not make a quantity a vector — the test is whether a direction is part of the answer, and for an energy it is not.
Speed and velocity have the same unit but are not the same quantity. A car going round a bend at a steady has constant speed and changing velocity, because its direction keeps changing. That distinction becomes the whole point of the acceleration section later on this page.
How do you calculate distance and displacement for a given path?
Add every part of the path for the distance; measure straight from start to finish for the displacement.
Worked example 1 — a right-angled path. Walk m east, then m north.
- Distance m
- Displacement: the straight line from start to finish is the hypotenuse of a right triangle,
directed north-east. So m of walking produced m of displacement.
Worked example 2 — going and coming back. Walk m east, then m west.
- Distance m
- Displacement m east
Taking east as positive, the two legs are m and m, and the signs do the subtraction for you.
Worked example 3 — a full lap. One lap of a m track: distance m, displacement .
Worked example 4 — half a lap. A circular track has radius m. Run from one end of a diameter to the other, along the curve. Taking :
- Distance m
- Displacement the diameter m
Worked example 5 — a there-and-back trip. A cyclist rides km north, then km east, then km south.
- Distance km
- Displacement: the north and south legs cancel, leaving km east
Displacement can never exceed distance. The shortest route between two points is the straight line, so the straight-line gap can at most equal the path length and is usually less. They are equal only for motion in a straight line with no reversal — which is exactly the case this chapter's equations of motion will assume.
Displacement can be zero while distance is large, but never the reverse. A zero distance means you never moved, which forces a zero displacement too. So the two quantities are not symmetric, and that asymmetry is worth stating in an answer rather than just tabulating the definitions.
Worked example 1 — a right-angled path. Walk m east, then m north.
- Distance m
- Displacement: the straight line from start to finish is the hypotenuse of a right triangle,
directed north-east. So m of walking produced m of displacement.
Worked example 2 — going and coming back. Walk m east, then m west.
- Distance m
- Displacement m east
Taking east as positive, the two legs are m and m, and the signs do the subtraction for you.
Worked example 3 — a full lap. One lap of a m track: distance m, displacement .
Worked example 4 — half a lap. A circular track has radius m. Run from one end of a diameter to the other, along the curve. Taking :
- Distance m
- Displacement the diameter m
Worked example 5 — a there-and-back trip. A cyclist rides km north, then km east, then km south.
- Distance km
- Displacement: the north and south legs cancel, leaving km east
Displacement can never exceed distance. The shortest route between two points is the straight line, so the straight-line gap can at most equal the path length and is usually less. They are equal only for motion in a straight line with no reversal — which is exactly the case this chapter's equations of motion will assume.
Displacement can be zero while distance is large, but never the reverse. A zero distance means you never moved, which forces a zero displacement too. So the two quantities are not symmetric, and that asymmetry is worth stating in an answer rather than just tabulating the definitions.
Formula
What is the formula for average speed and average velocity?
Average speed uses total distance; average velocity uses net displacement. Same total time on the bottom, different quantity on top.
Both are measured in . Average speed is a scalar and average velocity a vector.
Worked example 1 — straight-line motion. A car covers m in s and then a further m in s, all in the same direction.
The displacement is also m, so the average velocity is in that direction. The two agree because the motion never reversed.
Worked example 2 — with a return. A boy walks m east in s and then m west in s.
Moving for half a minute at an average velocity of zero is not a contradiction — it is what a round trip means.
Worked example 3 — two stretches at different speeds. A bus travels at for h and then at for h.
Worked example 4 — equal distances, not equal times. A car covers the first half of a journey at and the second half at . Let each half be km.
**Not .** The plain average of and is wrong because the car spends twice as long on the slow half as on the fast one, so the slow speed gets more weight. Averaging the two speeds is only valid when the two TIMES are equal, not the two distances — and in worked example 3, where the times were h and h, the answer likewise is not the plain average of and .
Unit conversion, since questions mix the two. To go from to , multiply by :
and to go back, multiply by .
Both are measured in . Average speed is a scalar and average velocity a vector.
Worked example 1 — straight-line motion. A car covers m in s and then a further m in s, all in the same direction.
The displacement is also m, so the average velocity is in that direction. The two agree because the motion never reversed.
Worked example 2 — with a return. A boy walks m east in s and then m west in s.
Moving for half a minute at an average velocity of zero is not a contradiction — it is what a round trip means.
Worked example 3 — two stretches at different speeds. A bus travels at for h and then at for h.
Worked example 4 — equal distances, not equal times. A car covers the first half of a journey at and the second half at . Let each half be km.
**Not .** The plain average of and is wrong because the car spends twice as long on the slow half as on the fast one, so the slow speed gets more weight. Averaging the two speeds is only valid when the two TIMES are equal, not the two distances — and in worked example 3, where the times were h and h, the answer likewise is not the plain average of and .
Unit conversion, since questions mix the two. To go from to , multiply by :
and to go back, multiply by .
How do you calculate acceleration, and when is it a retardation?
Acceleration is the change in velocity divided by the time taken, and it is called a retardation when it acts against the motion and slows the body down.
with the initial velocity, the final velocity, and the SI unit . Acceleration is a vector.
Worked example 1 — speeding up from rest. A car reaches from rest in s:
Worked example 2 — braking. A car at comes to rest in s:
The minus sign says the acceleration opposes the motion, so this is a **retardation of **. Reporting a *retardation of * says the same thing twice and is a marked error.
Worked example 3 — with unit conversion. A train speeds up from to in s. Convert first:
Worked example 4 — finding the final velocity. A cyclist at accelerates at for s:
Worked example 5 — finding the time. A scooter at retards at . Time to stop:
The sign convention. Choose one direction as positive and keep it for the whole problem. Then:
- and with the same sign — the body is speeding up
- and with opposite signs — the body is slowing down, a retardation
A negative acceleration does not always mean slowing down. A body moving in the negative direction with a negative acceleration is speeding up — going faster and faster the negative way. So retardation is about the relative signs of and , not about being negative on its own. That is the distinction the sign convention exists to make.
Uniform circular motion has an acceleration even at constant speed. A car going round a roundabout at a steady has unchanging speed and continuously changing velocity, because its direction keeps turning. Since acceleration is the rate of change of velocity, there is an acceleration throughout. A body with zero acceleration must be moving in a straight line at a steady speed — which is what the rest of this chapter assumes.
with the initial velocity, the final velocity, and the SI unit . Acceleration is a vector.
Worked example 1 — speeding up from rest. A car reaches from rest in s:
Worked example 2 — braking. A car at comes to rest in s:
The minus sign says the acceleration opposes the motion, so this is a **retardation of **. Reporting a *retardation of * says the same thing twice and is a marked error.
Worked example 3 — with unit conversion. A train speeds up from to in s. Convert first:
Worked example 4 — finding the final velocity. A cyclist at accelerates at for s:
Worked example 5 — finding the time. A scooter at retards at . Time to stop:
The sign convention. Choose one direction as positive and keep it for the whole problem. Then:
- and with the same sign — the body is speeding up
- and with opposite signs — the body is slowing down, a retardation
A negative acceleration does not always mean slowing down. A body moving in the negative direction with a negative acceleration is speeding up — going faster and faster the negative way. So retardation is about the relative signs of and , not about being negative on its own. That is the distinction the sign convention exists to make.
Uniform circular motion has an acceleration even at constant speed. A car going round a roundabout at a steady has unchanging speed and continuously changing velocity, because its direction keeps turning. Since acceleration is the rate of change of velocity, there is an acceleration throughout. A body with zero acceleration must be moving in a straight line at a steady speed — which is what the rest of this chapter assumes.
Exam tip
Exam tip: fix a positive direction before writing a single number
Write "taking east as positive" (or upward, or the direction of motion) before the working. Every sign that follows depends on it, and an unstated convention loses interpretation marks.
Convert units first. Multiply by to reach : becomes , becomes , becomes .
Distance is the whole path; displacement is start to finish. A full lap gives m and .
Displacement never exceeds distance, and the two are equal only for straight-line motion with no reversal.
Average speed uses distance, average velocity uses displacement — both over the total time.
Never average two speeds directly unless the two times are equal. Equal distances at and give , not .
Retardation is the magnitude: write *a retardation of , never a retardation of *.
Give the unit every time — for speed, for acceleration — and state the direction for any vector answer.
Label a vector diagram with its scale and an arrowhead: N at cm N is a cm arrow.
Work and energy are scalars even though force is a vector.
And remember constant speed round a bend still means an acceleration, because the direction is changing.
Convert units first. Multiply by to reach : becomes , becomes , becomes .
Distance is the whole path; displacement is start to finish. A full lap gives m and .
Displacement never exceeds distance, and the two are equal only for straight-line motion with no reversal.
Average speed uses distance, average velocity uses displacement — both over the total time.
Never average two speeds directly unless the two times are equal. Equal distances at and give , not .
Retardation is the magnitude: write *a retardation of , never a retardation of *.
Give the unit every time — for speed, for acceleration — and state the direction for any vector answer.
Label a vector diagram with its scale and an arrowhead: N at cm N is a cm arrow.
Work and energy are scalars even though force is a vector.
And remember constant speed round a bend still means an acceleration, because the direction is changing.
Did you know
Why a car can have a constant speed and never a constant velocity
Drive a car round a perfectly circular track, holding the speedometer at exactly — that is — for the whole trip.
The speed never changes. Not once, not by a fraction.
And the velocity changes at every single instant, because the direction is turning continuously. A vector is only unchanged when both its size and its direction stay put, and one of the two is in constant motion here.
So the car is accelerating for the entire trip, even though the driver never touches the accelerator pedal differently. Turning the steering wheel is accelerating, in the physics sense.
That is not a verbal trick. You can feel it: passengers get pushed towards the outside of the bend, and the tyres have to grip the road sideways to supply the force. No acceleration would mean no force and no sideways push, and a car with no sideways grip on a bend goes straight on — which is the whole reason a wet road is dangerous at a corner.
And the displacement behaves just as oddly. After one complete circuit the car has travelled the full circumference and has a displacement of zero, so its average velocity for the lap is zero while its average speed was throughout.
All three oddities come from the same source: **speed, distance and the ordinary word fast ignore direction, and velocity, displacement and acceleration do not.** Once a direction is part of the quantity, going round in a circle stops being a way of getting nowhere slowly and becomes a way of getting nowhere while accelerating the whole time.
The speed never changes. Not once, not by a fraction.
And the velocity changes at every single instant, because the direction is turning continuously. A vector is only unchanged when both its size and its direction stay put, and one of the two is in constant motion here.
So the car is accelerating for the entire trip, even though the driver never touches the accelerator pedal differently. Turning the steering wheel is accelerating, in the physics sense.
That is not a verbal trick. You can feel it: passengers get pushed towards the outside of the bend, and the tyres have to grip the road sideways to supply the force. No acceleration would mean no force and no sideways push, and a car with no sideways grip on a bend goes straight on — which is the whole reason a wet road is dangerous at a corner.
And the displacement behaves just as oddly. After one complete circuit the car has travelled the full circumference and has a displacement of zero, so its average velocity for the lap is zero while its average speed was throughout.
All three oddities come from the same source: **speed, distance and the ordinary word fast ignore direction, and velocity, displacement and acceleration do not.** Once a direction is part of the quantity, going round in a circle stops being a way of getting nowhere slowly and becomes a way of getting nowhere while accelerating the whole time.
Exam relevance
How do scalars, vectors and acceleration feed into JEE Main and NEET?
Because vector handling is the tool used in every mechanics chapter, and the distinction between average and instantaneous quantities is where calculus enters physics.
This is the foundation for Class 11 Physics Motion in a Straight Line and Motion in a Plane, examined in JEE Main and NEET. The first of those repeats this page's content with one addition: instantaneous velocity and acceleration, defined as the limits
where this page uses averages over a finite interval. The average-against-instantaneous distinction is the whole reason the derivative appears in physics, and questions contrasting the two are a standard type.
Vector addition is the bigger extension. Class 11 Motion in a Plane adds vectors by components and by the parallelogram and triangle laws, and the right-angled walk on this page — m east then m north giving m — is the simplest case of it. Resolving a vector into components becomes routine there and stays routine through Laws of Motion and Work, Energy and Power.
The circular-motion observation is where this leads directly. Class 11 shows that a body moving in a circle at constant speed has an acceleration of magnitude directed towards the centre — centripetal acceleration — and the reason it exists at all is the point made on this page, that changing direction changes velocity. Numericals on it are recurring JEE Main material, and the conceptual version appears in NEET as an assertion-reason item.
Relative velocity is the other standard extension, and it depends on treating velocity as a signed quantity in one dimension exactly as done here. River-boat and rain-umbrella problems in Class 11 are vector subtractions.
Where the averaging trap reappears. The equal-distance result — and giving , not — is the harmonic mean, and questions built on it appear in both board and competitive papers precisely because the plain average looks right. The companion result, that equal times do give the plain average, is the pair to know.
What the questions look like. For board work, expect classify quantities as scalar or vector, find distance and displacement for a described path, average speed and average velocity with a return leg, acceleration from a change in velocity with unit conversion, and a vector diagram to scale. For JEE Main and NEET, expect graph-based and calculus-based kinematics, vector components, relative velocity, and centripetal acceleration.
How board and competitive emphasis differ. A board paper rewards the stated sign convention, the units and the direction on every vector answer. A competitive paper assumes all of that and tests whether you can work with components and instantaneous rates.
The single trap that costs the most marks. Averaging two speeds when the distances are equal rather than the times. A car covering equal halves at and averages , because it spends twice as long going slowly. The defence is to compute the total distance and the total time separately every time and only then divide — never to average the speeds themselves.
This is the foundation for Class 11 Physics Motion in a Straight Line and Motion in a Plane, examined in JEE Main and NEET. The first of those repeats this page's content with one addition: instantaneous velocity and acceleration, defined as the limits
where this page uses averages over a finite interval. The average-against-instantaneous distinction is the whole reason the derivative appears in physics, and questions contrasting the two are a standard type.
Vector addition is the bigger extension. Class 11 Motion in a Plane adds vectors by components and by the parallelogram and triangle laws, and the right-angled walk on this page — m east then m north giving m — is the simplest case of it. Resolving a vector into components becomes routine there and stays routine through Laws of Motion and Work, Energy and Power.
The circular-motion observation is where this leads directly. Class 11 shows that a body moving in a circle at constant speed has an acceleration of magnitude directed towards the centre — centripetal acceleration — and the reason it exists at all is the point made on this page, that changing direction changes velocity. Numericals on it are recurring JEE Main material, and the conceptual version appears in NEET as an assertion-reason item.
Relative velocity is the other standard extension, and it depends on treating velocity as a signed quantity in one dimension exactly as done here. River-boat and rain-umbrella problems in Class 11 are vector subtractions.
Where the averaging trap reappears. The equal-distance result — and giving , not — is the harmonic mean, and questions built on it appear in both board and competitive papers precisely because the plain average looks right. The companion result, that equal times do give the plain average, is the pair to know.
What the questions look like. For board work, expect classify quantities as scalar or vector, find distance and displacement for a described path, average speed and average velocity with a return leg, acceleration from a change in velocity with unit conversion, and a vector diagram to scale. For JEE Main and NEET, expect graph-based and calculus-based kinematics, vector components, relative velocity, and centripetal acceleration.
How board and competitive emphasis differ. A board paper rewards the stated sign convention, the units and the direction on every vector answer. A competitive paper assumes all of that and tests whether you can work with components and instantaneous rates.
The single trap that costs the most marks. Averaging two speeds when the distances are equal rather than the times. A car covering equal halves at and averages , because it spends twice as long going slowly. The defence is to compute the total distance and the total time separately every time and only then divide — never to average the speeds themselves.
Key takeaways
Scalars, vectors, displacement and acceleration: quick revision
- A scalar needs only a magnitude; a vector needs a magnitude and a direction.
- Scalars: mass, distance, speed, time, volume, density, area, work, energy, power, temperature, pressure.
- Vectors: displacement, velocity, acceleration, force, weight, momentum.
- Work and energy are scalars even though force is a vector — being built from a vector is not the test.
- A vector is drawn as a directed line segment with a stated scale, the right length and an arrowhead: N at cm N is a cm arrow.
- Distance is the whole path; displacement is the straight line from start to finish, with a direction.
- m east then m north: distance m, displacement m north-east.
- m east then m west: distance m, displacement m east.
- One lap of a m track: distance m, displacement zero.
- Half a circular track of radius m: distance m, displacement m.
- km north, km east, km south: distance km, displacement km east.
- Displacement never exceeds distance, and they are equal only in straight-line motion without reversal.
- Average speed ; average velocity .
- m in s then m in s straight: both come to .
- m east in s then m west in s: average speed , average velocity zero.
- for h then for h: .
- **Equal distances at and give , not — the slow half takes twice as long.
- Convert with **: ; ; .
- Acceleration , in , a vector.
- Rest to in s gives ; to rest in s gives , a **retardation of **.
- to in s gives ; at for s reaches ; retarding at stops in s.
- **Same signs for and mean speeding up; opposite signs mean slowing down** — a negative alone does not mean retardation.
- Constant speed round a bend is still an acceleration, because the direction changes.
Time yourself walking to a shop and back, work out both your average speed and your average velocity for the round trip, and see which one your legs agree with.
- Scalars: mass, distance, speed, time, volume, density, area, work, energy, power, temperature, pressure.
- Vectors: displacement, velocity, acceleration, force, weight, momentum.
- Work and energy are scalars even though force is a vector — being built from a vector is not the test.
- A vector is drawn as a directed line segment with a stated scale, the right length and an arrowhead: N at cm N is a cm arrow.
- Distance is the whole path; displacement is the straight line from start to finish, with a direction.
- m east then m north: distance m, displacement m north-east.
- m east then m west: distance m, displacement m east.
- One lap of a m track: distance m, displacement zero.
- Half a circular track of radius m: distance m, displacement m.
- km north, km east, km south: distance km, displacement km east.
- Displacement never exceeds distance, and they are equal only in straight-line motion without reversal.
- Average speed ; average velocity .
- m in s then m in s straight: both come to .
- m east in s then m west in s: average speed , average velocity zero.
- for h then for h: .
- **Equal distances at and give , not — the slow half takes twice as long.
- Convert with **: ; ; .
- Acceleration , in , a vector.
- Rest to in s gives ; to rest in s gives , a **retardation of **.
- to in s gives ; at for s reaches ; retarding at stops in s.
- **Same signs for and mean speeding up; opposite signs mean slowing down** — a negative alone does not mean retardation.
- Constant speed round a bend is still an acceleration, because the direction changes.
Time yourself walking to a shop and back, work out both your average speed and your average velocity for the round trip, and see which one your legs agree with.