A Graph of Motion Hides Two Answers: One in the Slope, One in the Area
Learn to plot position-time graphs for rest, uniform and non-uniform motion, read velocity from a slope, interpret velocity-time graphs, and find displacement from the area beneath one.
What do the slope and the area of a motion graph tell you?
Two different things, and which is which depends on what is plotted up the vertical axis.
On a position-time graph, the slope is the velocity. Rise divided by run is metres divided by seconds, so the slope has the units of velocity and no other meaning is available to it.
On a velocity-time graph, the slope is the acceleration, because metres per second divided by seconds gives . And on the same graph the area underneath is the displacement, because metres per second multiplied by seconds gives metres.
So a single velocity-time graph carries the acceleration in its steepness and the displacement in the space beneath it. Nothing has to be remembered by rote here — check the units of rise over run, or of height times width, and the meaning follows. This page covers the second part of the CBSE Class 9 Science chapter on describing motion.
On a position-time graph, the slope is the velocity. Rise divided by run is metres divided by seconds, so the slope has the units of velocity and no other meaning is available to it.
On a velocity-time graph, the slope is the acceleration, because metres per second divided by seconds gives . And on the same graph the area underneath is the displacement, because metres per second multiplied by seconds gives metres.
So a single velocity-time graph carries the acceleration in its steepness and the displacement in the space beneath it. Nothing has to be remembered by rote here — check the units of rise over run, or of height times width, and the meaning follows. This page covers the second part of the CBSE Class 9 Science chapter on describing motion.
How do you plot a position-time graph from a table of data?
Time on the horizontal axis, position on the vertical axis, one point per row, then join them.
The shape of the line tells you the kind of motion at a glance.
An object at rest. Its position does not change, so every point has the same height and the graph is a horizontal straight line. Time passes; position does not move.
Uniform motion. Equal distances in equal times, so the graph is a straight sloping line.
Take this table:
- s, m
- s, m
- s, m
- s, m
- s, m
The position rises by the same m in every second, so the points lie on a straight line — uniform motion at m/s.
Non-uniform motion. Unequal distances in equal times, so the graph is a curve.
- s, m
- s, m
- s, m
- s, m
The gaps are m, m and m — increasing, so the object is speeding up and the graph curves upward, getting steeper as it goes.
Everyday evidence. A train moving steadily between two stations traces a straight line on such a graph. A train pulling out of a station traces a curve that starts almost flat and steepens.
A position-time graph can never be vertical. A vertical segment would mean the object was in two places at the same instant, which is impossible — and would also mean infinite velocity. So however steep a real graph gets, it leans; it never stands up straight. That single impossibility is a useful check when a question asks which of four sketched graphs is not physically possible.
Label the axes with quantity and unit. Time (s) and Position (m), with the scale marked. An unlabelled graph cannot be read, and the labels carry marks of their own.
The shape of the line tells you the kind of motion at a glance.
An object at rest. Its position does not change, so every point has the same height and the graph is a horizontal straight line. Time passes; position does not move.
Uniform motion. Equal distances in equal times, so the graph is a straight sloping line.
Take this table:
- s, m
- s, m
- s, m
- s, m
- s, m
The position rises by the same m in every second, so the points lie on a straight line — uniform motion at m/s.
Non-uniform motion. Unequal distances in equal times, so the graph is a curve.
- s, m
- s, m
- s, m
- s, m
The gaps are m, m and m — increasing, so the object is speeding up and the graph curves upward, getting steeper as it goes.
Everyday evidence. A train moving steadily between two stations traces a straight line on such a graph. A train pulling out of a station traces a curve that starts almost flat and steepens.
A position-time graph can never be vertical. A vertical segment would mean the object was in two places at the same instant, which is impossible — and would also mean infinite velocity. So however steep a real graph gets, it leans; it never stands up straight. That single impossibility is a useful check when a question asks which of four sketched graphs is not physically possible.
Label the axes with quantity and unit. Time (s) and Position (m), with the scale marked. An unlabelled graph cannot be read, and the labels carry marks of their own.
How do you find velocity from the slope of a position-time graph?
Take two points on the line and divide the change in position by the change in time.
Worked example 1. On the uniform-motion graph above, take the points and .
Any other pair of points on the same straight line gives the same answer, which is what uniform means.
Worked example 2. A graph runs from to .
Worked example 3 — a negative slope. An object's position falls from m at to m at s.
The negative sign means the object is moving in the negative direction — back towards the origin — at m/s.
Worked example 4 — a horizontal line. If the position stays at m from s to s, the slope is , so the object is at rest during that interval.
Steeper means faster. Two objects on the same axes can be compared at a glance: the one whose line is steeper has the greater speed. This is why a single pair of axes is used to compare journeys.
Take the two points far apart. Choosing points close together makes small reading errors matter far more, because you are dividing two small numbers. Using and is more reliable than using and , even though both give the same true answer.
For a curve, the slope changes from point to point. A curved position-time graph has a different velocity at every instant, so asking for the velocity is meaningless — you can only give the velocity at a stated time, found from the steepness there, or the average velocity over an interval, found from the straight line joining its two ends.
Worked example 1. On the uniform-motion graph above, take the points and .
Any other pair of points on the same straight line gives the same answer, which is what uniform means.
Worked example 2. A graph runs from to .
Worked example 3 — a negative slope. An object's position falls from m at to m at s.
The negative sign means the object is moving in the negative direction — back towards the origin — at m/s.
Worked example 4 — a horizontal line. If the position stays at m from s to s, the slope is , so the object is at rest during that interval.
Steeper means faster. Two objects on the same axes can be compared at a glance: the one whose line is steeper has the greater speed. This is why a single pair of axes is used to compare journeys.
Take the two points far apart. Choosing points close together makes small reading errors matter far more, because you are dividing two small numbers. Using and is more reliable than using and , even though both give the same true answer.
For a curve, the slope changes from point to point. A curved position-time graph has a different velocity at every instant, so asking for the velocity is meaningless — you can only give the velocity at a stated time, found from the steepness there, or the average velocity over an interval, found from the straight line joining its two ends.
How do you read acceleration from a velocity-time graph?
Velocity is now on the vertical axis, so the slope is the acceleration.
The shape again tells the story.
A horizontal line — velocity constant, so zero acceleration. The object moves at a steady speed in a fixed direction.
A straight sloping line — velocity changing at a steady rate, so uniform acceleration, and the slope gives its value.
A curve — the rate of change of velocity is itself changing, so the acceleration is non-uniform.
Worked example 1. Velocity rises from to m/s over s along a straight line.
Worked example 2 — deceleration. Velocity falls from m/s to m/s over s.
A downward sloping line means the acceleration is negative — the object is slowing.
Worked example 3 — a line not starting at zero. Velocity rises from m/s at to m/s at s.
Everyday evidence. A car pulling away from a traffic light shows a line climbing steeply at first and then flattening as it approaches cruising speed — non-uniform acceleration, falling towards zero. Once it is cruising, the line is horizontal.
A horizontal line does not mean the object has stopped. It means the velocity is not changing. A horizontal line at m/s describes a car travelling steadily at m/s; only a horizontal line at zero describes an object at rest. Confusing flat graph with not moving is the single commonest misreading of a velocity-time graph, and it comes from carrying over the rule for a position-time graph, where a flat line does mean at rest.
So the same shape means different things on the two graphs. A horizontal line means at rest on a position-time graph and constant velocity on a velocity-time graph. Checking which quantity is on the vertical axis before interpreting anything is not a formality — it changes the answer.
The shape again tells the story.
A horizontal line — velocity constant, so zero acceleration. The object moves at a steady speed in a fixed direction.
A straight sloping line — velocity changing at a steady rate, so uniform acceleration, and the slope gives its value.
A curve — the rate of change of velocity is itself changing, so the acceleration is non-uniform.
Worked example 1. Velocity rises from to m/s over s along a straight line.
Worked example 2 — deceleration. Velocity falls from m/s to m/s over s.
A downward sloping line means the acceleration is negative — the object is slowing.
Worked example 3 — a line not starting at zero. Velocity rises from m/s at to m/s at s.
Everyday evidence. A car pulling away from a traffic light shows a line climbing steeply at first and then flattening as it approaches cruising speed — non-uniform acceleration, falling towards zero. Once it is cruising, the line is horizontal.
A horizontal line does not mean the object has stopped. It means the velocity is not changing. A horizontal line at m/s describes a car travelling steadily at m/s; only a horizontal line at zero describes an object at rest. Confusing flat graph with not moving is the single commonest misreading of a velocity-time graph, and it comes from carrying over the rule for a position-time graph, where a flat line does mean at rest.
So the same shape means different things on the two graphs. A horizontal line means at rest on a position-time graph and constant velocity on a velocity-time graph. Checking which quantity is on the vertical axis before interpreting anything is not a formality — it changes the answer.
How do you get displacement from the area under a velocity-time graph?
Find the area between the line and the time axis, splitting it into rectangles, triangles and trapeziums.
The reason is the units: velocity in m/s multiplied by time in s gives metres. So area is displacement, for any shape of graph.
Worked example 1 — constant velocity. An object moves at m/s for s. The graph is a horizontal line, and the area beneath it is a rectangle:
Worked example 2 — uniform acceleration from rest. Velocity rises from to m/s in s. The area is a triangle:
Check it another way. The average velocity over that interval is m/s, and m. The two methods agree, as they must.
Worked example 3 — a full journey in three phases. A bus accelerates from rest to m/s in s, travels at m/s for s, then decelerates uniformly to rest in s.
The graph is a trapezium, and it is easiest split into three pieces:
- Triangle:
- Rectangle:
- Triangle:
The total time is s, so
which is less than the top speed of m/s, as it should be.
Check by the trapezium formula. The whole area is a trapezium with parallel sides s and s and height m/s:
The same answer in one line.
Area below the time axis counts as negative. If a body moves forward at m/s for s and then backward at m/s for s, the two areas are m and m:
So the signed area gives displacement and the total area gives distance — exactly the distinction from the first part of this chapter, now visible as a picture.
Do not read the area as a velocity, or the slope as a displacement. Both are common slips, and both are caught by checking units: rise over run gives here, and height times width gives metres. The units decide which is which, and no memorised rule is needed.
The reason is the units: velocity in m/s multiplied by time in s gives metres. So area is displacement, for any shape of graph.
Worked example 1 — constant velocity. An object moves at m/s for s. The graph is a horizontal line, and the area beneath it is a rectangle:
Worked example 2 — uniform acceleration from rest. Velocity rises from to m/s in s. The area is a triangle:
Check it another way. The average velocity over that interval is m/s, and m. The two methods agree, as they must.
Worked example 3 — a full journey in three phases. A bus accelerates from rest to m/s in s, travels at m/s for s, then decelerates uniformly to rest in s.
The graph is a trapezium, and it is easiest split into three pieces:
- Triangle:
- Rectangle:
- Triangle:
The total time is s, so
which is less than the top speed of m/s, as it should be.
Check by the trapezium formula. The whole area is a trapezium with parallel sides s and s and height m/s:
The same answer in one line.
Area below the time axis counts as negative. If a body moves forward at m/s for s and then backward at m/s for s, the two areas are m and m:
So the signed area gives displacement and the total area gives distance — exactly the distinction from the first part of this chapter, now visible as a picture.
Do not read the area as a velocity, or the slope as a displacement. Both are common slips, and both are caught by checking units: rise over run gives here, and height times width gives metres. The units decide which is which, and no memorised rule is needed.
Exam tip
Exam tip: check which quantity is on the vertical axis first
Look at the vertical axis before interpreting anything. A horizontal line means at rest on a position-time graph but constant velocity on a velocity-time graph.
Position-time graph: slope is velocity. Straight sloping line means uniform motion, curve means non-uniform, horizontal means at rest. A vertical line is impossible.
Velocity-time graph: slope is acceleration, and the area is displacement.
Check with units rather than memory: is velocity, is , and is metres.
When finding a slope, take two points far apart so reading errors matter less.
A negative slope means motion in the negative direction (position-time) or slowing down (velocity-time).
For an area, split it into rectangles and triangles, or use the trapezium formula in one step — the three-phase journey gives m either way.
Area below the axis is negative: signed area gives displacement, total area gives distance.
Label both axes with quantity and unit, mark the scale, and plot points neatly — graph marks are awarded for the drawing as well as the answer.
And on a curve, there is no single velocity — give it at a stated time, or give the average over an interval.
Position-time graph: slope is velocity. Straight sloping line means uniform motion, curve means non-uniform, horizontal means at rest. A vertical line is impossible.
Velocity-time graph: slope is acceleration, and the area is displacement.
Check with units rather than memory: is velocity, is , and is metres.
When finding a slope, take two points far apart so reading errors matter less.
A negative slope means motion in the negative direction (position-time) or slowing down (velocity-time).
For an area, split it into rectangles and triangles, or use the trapezium formula in one step — the three-phase journey gives m either way.
Area below the axis is negative: signed area gives displacement, total area gives distance.
Label both axes with quantity and unit, mark the scale, and plot points neatly — graph marks are awarded for the drawing as well as the answer.
And on a curve, there is no single velocity — give it at a stated time, or give the average over an interval.
Did you know
Why a graph answers questions the numbers do not
A table of readings and a graph of the same readings contain identical information, and yet some questions are almost impossible to answer from one and obvious from the other.
Given a column of twenty position readings, try to say whether the object was speeding up or slowing down. It means working out nineteen differences and comparing them — possible, tedious, and easy to get wrong. Now look at the same data plotted, and the answer is the direction in which the line bends. No arithmetic at all.
Try to find where two objects were at the same place at the same time. In two tables it means hunting for matching pairs. On one pair of axes it is the point where the lines cross.
Ask when an object was momentarily at rest. In a table you look for two equal consecutive readings and hope you did not miss the instant between them. On a graph it is where the line is flat.
The graph does this by turning a relationship into a shape, and shapes are something the eye reads in a single glance. It is the same reason a route map is easier to use than a list of station names with distances — the information is the same, and one form matches how we see.
There is a second gain, less obvious. A graph makes a wrong reading visible. One point that sits well off an otherwise straight line announces itself immediately, where the same error buried in a column of figures could pass unnoticed through an entire calculation.
So plotting is not a presentational step performed after the physics is finished. On a graph the slope, the area, the crossings and the flat parts are each a physical quantity you can point at — which is why this chapter spends a whole page on drawing before the next one reaches the equations.
Given a column of twenty position readings, try to say whether the object was speeding up or slowing down. It means working out nineteen differences and comparing them — possible, tedious, and easy to get wrong. Now look at the same data plotted, and the answer is the direction in which the line bends. No arithmetic at all.
Try to find where two objects were at the same place at the same time. In two tables it means hunting for matching pairs. On one pair of axes it is the point where the lines cross.
Ask when an object was momentarily at rest. In a table you look for two equal consecutive readings and hope you did not miss the instant between them. On a graph it is where the line is flat.
The graph does this by turning a relationship into a shape, and shapes are something the eye reads in a single glance. It is the same reason a route map is easier to use than a list of station names with distances — the information is the same, and one form matches how we see.
There is a second gain, less obvious. A graph makes a wrong reading visible. One point that sits well off an otherwise straight line announces itself immediately, where the same error buried in a column of figures could pass unnoticed through an entire calculation.
So plotting is not a presentational step performed after the physics is finished. On a graph the slope, the area, the crossings and the flat parts are each a physical quantity you can point at — which is why this chapter spends a whole page on drawing before the next one reaches the equations.
Exam relevance
Why are graph questions so common in JEE Main?
Because a graph can test three ideas at once — slope, area and sign — and a candidate who has only memorised formulas cannot get through it.
This page is the foundation for the Class 11 Physics chapter Motion in a Straight Line, examined in JEE Main and in NEET Physics. That chapter treats the same two graph types formally, adds the acceleration-time graph, and replaces the slope with the derivative and the area with the integral — but the physical meanings stay exactly as they are here. Slope of position-time is velocity; slope of velocity-time is acceleration; area under velocity-time is displacement. Those three statements survive unchanged into calculus.
Where it is reused. Class 11 Motion in a Plane applies the same graph reading to each component separately. Laws of Motion uses velocity-time graphs to show the effect of a force acting for a time. And in Class 12, the identical slope-and-area reasoning reappears for charge against time in Current Electricity and for other physical pairs — which is why the units-based check taught above is worth more than a memorised list.
What the questions look like. Graph-to-graph matching is the classic form: given a velocity-time graph, choose the corresponding position-time or acceleration-time graph. Numericals ask for the displacement in a named interval, or the acceleration during one phase of a multi-phase journey. Assertion-reason items are frequent, and a standard one pairs the area under a velocity-time graph gives displacement with a reason about units or about signed areas. Which of these graphs is impossible items exploit the vertical-line argument made above.
How board and competitive emphasis differ. A board paper asks you to plot a graph from a table and then read one quantity from it, and gives marks for axes, scale and neatness. A competitive paper gives you the graph already drawn and asks for something that requires two steps — the displacement in the final phase only, or the average velocity of the whole journey when the phases have different accelerations. Board work rewards drawing; competitive work rewards reading.
The single trap that costs the most marks. Reading a horizontal line on a velocity-time graph as an object at rest. It means constant velocity, and only a horizontal line at zero means at rest. The error comes from carrying over the position-time rule, and the fix is the habit above: check the vertical axis first, every time.
A second trap worth naming. Ignoring the sign of the area in a journey that reverses. Total area gives distance and signed area gives displacement, and a question asking for displacement on a graph that dips below the axis is testing exactly that — the same distance-and-displacement distinction from the first part of this chapter, now presented as a picture rather than as a description in words.
This page is the foundation for the Class 11 Physics chapter Motion in a Straight Line, examined in JEE Main and in NEET Physics. That chapter treats the same two graph types formally, adds the acceleration-time graph, and replaces the slope with the derivative and the area with the integral — but the physical meanings stay exactly as they are here. Slope of position-time is velocity; slope of velocity-time is acceleration; area under velocity-time is displacement. Those three statements survive unchanged into calculus.
Where it is reused. Class 11 Motion in a Plane applies the same graph reading to each component separately. Laws of Motion uses velocity-time graphs to show the effect of a force acting for a time. And in Class 12, the identical slope-and-area reasoning reappears for charge against time in Current Electricity and for other physical pairs — which is why the units-based check taught above is worth more than a memorised list.
What the questions look like. Graph-to-graph matching is the classic form: given a velocity-time graph, choose the corresponding position-time or acceleration-time graph. Numericals ask for the displacement in a named interval, or the acceleration during one phase of a multi-phase journey. Assertion-reason items are frequent, and a standard one pairs the area under a velocity-time graph gives displacement with a reason about units or about signed areas. Which of these graphs is impossible items exploit the vertical-line argument made above.
How board and competitive emphasis differ. A board paper asks you to plot a graph from a table and then read one quantity from it, and gives marks for axes, scale and neatness. A competitive paper gives you the graph already drawn and asks for something that requires two steps — the displacement in the final phase only, or the average velocity of the whole journey when the phases have different accelerations. Board work rewards drawing; competitive work rewards reading.
The single trap that costs the most marks. Reading a horizontal line on a velocity-time graph as an object at rest. It means constant velocity, and only a horizontal line at zero means at rest. The error comes from carrying over the position-time rule, and the fix is the habit above: check the vertical axis first, every time.
A second trap worth naming. Ignoring the sign of the area in a journey that reverses. Total area gives distance and signed area gives displacement, and a question asking for displacement on a graph that dips below the axis is testing exactly that — the same distance-and-displacement distinction from the first part of this chapter, now presented as a picture rather than as a description in words.
Key takeaways
Motion graphs: quick revision
- Position-time graph: slope is velocity. Horizontal means at rest, straight sloping means uniform motion, curve means non-uniform.
- A table of m at s is a straight line — uniform motion at m/s.
- A table of m has gaps of m, so it curves upward — speeding up.
- A position-time graph can never be vertical — that would mean being in two places at once.
- Slope . From to gives m/s; from to gives m/s.
- A negative slope means motion in the negative direction: m falling to m in s gives m/s.
- Take two points far apart to reduce reading error, and on a curve give the velocity at a stated time or the average over an interval.
- Velocity-time graph: slope is acceleration, area is displacement.
- Horizontal means zero acceleration (constant velocity, not at rest); straight sloping means uniform acceleration; curve means non-uniform.
- to m/s in s gives ; to m/s in s gives ; to m/s in s gives .
- Area under a velocity-time graph is displacement, because m/s s m.
- m/s for s gives a rectangle of m. to m/s in s gives a triangle of m, matching average velocity m/s s.
- Three-phase journey: m, or in one step m. Average velocity m/s.
- Area below the axis is negative: m then m gives displacement and distance m.
- Check the vertical axis before interpreting, and use units to decide whether slope or area is wanted.
- Label axes with quantity and unit, and mark the scale.
Sketch the velocity-time graph of your own journey to school from memory, then work out the displacement from its area and compare it with the distance you know — the gap between the two will tell you how many turns the route has.
- A table of m at s is a straight line — uniform motion at m/s.
- A table of m has gaps of m, so it curves upward — speeding up.
- A position-time graph can never be vertical — that would mean being in two places at once.
- Slope . From to gives m/s; from to gives m/s.
- A negative slope means motion in the negative direction: m falling to m in s gives m/s.
- Take two points far apart to reduce reading error, and on a curve give the velocity at a stated time or the average over an interval.
- Velocity-time graph: slope is acceleration, area is displacement.
- Horizontal means zero acceleration (constant velocity, not at rest); straight sloping means uniform acceleration; curve means non-uniform.
- to m/s in s gives ; to m/s in s gives ; to m/s in s gives .
- Area under a velocity-time graph is displacement, because m/s s m.
- m/s for s gives a rectangle of m. to m/s in s gives a triangle of m, matching average velocity m/s s.
- Three-phase journey: m, or in one step m. Average velocity m/s.
- Area below the axis is negative: m then m gives displacement and distance m.
- Check the vertical axis before interpreting, and use units to decide whether slope or area is wanted.
- Label axes with quantity and unit, and mark the scale.
Sketch the velocity-time graph of your own journey to school from memory, then work out the displacement from its area and compare it with the distance you know — the gap between the two will tell you how many turns the route has.