Displacement-Time Graphs
A displacement-time graph represents how an object's position relative to a chosen origin changes over time. Its central idea is that the gradient of the graph gives velocity, so the shape of the graph reveals whether the object is stationary, moving forward, moving backward, or changing velocity. Understanding these graphs is fundamental in kinematics because they connect geometric features of a graph to physical motion and provide a visual way to interpret and calculate movement.
Displacement-time graph: A displacement-time graph shows displacement on the vertical axis against time on the horizontal axis. Displacement measures position relative to a fixed origin and includes direction, which is why values can be positive, zero, or negative depending on which side of the origin the object is on.
Displacement vs distance: Displacement is a vector quantity, so it can be negative if the object is on the opposite side of the chosen origin from the positive direction. Distance is a scalar quantity and cannot be negative, so a displacement-time graph can go below the time axis while a distance-time graph cannot.
Origin and sign convention: The value of displacement at any time tells you where the object is relative to the chosen reference point. A point above the axis means positive displacement, a point on the axis means the object is at the origin, and a point below the axis means negative displacement.
Reading a point on the graph: Each coordinate means that at time , the displacement is . This allows you to interpret not just how far the object is from the origin, but also whether it has crossed the origin or changed direction during its motion.
Displacement: If displacement is written as , then tells you the signed position of the object relative to the origin. This is the primary output of the graph and should always be interpreted together with its units, such as metres.
Time: Time is usually written as and is always plotted on the horizontal axis. Because motion is tracked as time progresses, you read the graph from left to right to describe the journey.
Velocity from graph shape: The most important interpretive feature is that the graph's gradient corresponds to velocity. A graph with different slopes at different times represents motion with changing velocity.
Key idea: The slope of the graph tells you how quickly displacement is changing, and therefore how fast and in which direction the object is moving.
Positive, negative, and zero gradients: A positive gradient means displacement increases with time, so the object moves in the positive direction. A negative gradient means displacement decreases with time, so the object moves in the negative direction, while a zero gradient means displacement is constant and the object is stationary.
Steepness and speed: The steeper the graph, the larger the magnitude of the gradient, and so the larger the speed. This works because speed is the magnitude of velocity, so both a steep positive line and a steep negative line indicate fast motion, though in opposite directions.
Straight versus curved graphs: A straight line has constant gradient, so it represents constant velocity. A curved line has changing gradient, which means the velocity is changing with time and the object is accelerating or decelerating.
Tangent interpretation: For a curved displacement-time graph, the velocity at a single instant is found from the gradient of the tangent at that point. This matters because the slope between two distant points only gives an average velocity, not the exact velocity at one moment.
Why curvature matters: If the curve becomes steeper over time, the magnitude of velocity is increasing. If the curve becomes less steep, the magnitude of velocity is decreasing, though the direction still depends on whether the slope is positive or negative.
How to read motion from the graph: Start by identifying the axes and units, because displacement-time graphs can be confused with distance-time or velocity-time graphs. Then read individual points to determine where the object is at a given time, and read the slope to determine how it is moving.
Finding average velocity between two times: Choose two points on the graph and compute the gradient using where and are the displacements at times and . This gives the net rate of change of displacement over that time interval, so direction is built into the sign of the answer.
Finding instantaneous velocity: If the graph is curved, draw or imagine a tangent at the required point and calculate its gradient. This works because the tangent gives the local rate of change of displacement at that instant.
Stationary intervals: Look for horizontal parts of the graph. Since displacement does not change during such intervals, the velocity is zero and the object is at rest.
Moving away from the origin or toward it: If the displacement value is increasing in the positive direction, the object is moving farther into positive position. If the graph moves toward the horizontal axis, the object is approaching the origin, and if it crosses the axis, the object passes through the origin.
Changing direction: A change in direction is shown when the gradient changes sign from positive to negative or from negative to positive. This matters because the object may still be far from the origin even while reversing its motion.
Average speed uses total distance travelled divided by total time: This value is always non-negative because distance ignores direction.
Average velocity uses net displacement divided by total time: This value can be positive, zero, or negative because displacement includes direction.
When each is needed: Use average speed when the question asks how fast the object travelled overall, regardless of direction. Use average velocity when the question asks for the net change in position per unit time, especially if the journey includes reversing direction.
This distinction matters because a negative value or negative slope has physical meaning on a displacement-time graph but not on a distance-time graph. Confusing the two often leads to incorrect conclusions about direction and sign.
A point shows position, not speed: The vertical coordinate tells you where the object is, but not how fast it is moving. Two objects can have the same displacement at a given time yet have different velocities if their graphs have different slopes there.
Crossing the axis vs stopping: If the graph crosses the time axis, the object is at the origin at that instant. This does not mean it is stationary; it is stationary only if the gradient is zero at that instant.
Above or below the axis: Being above the axis means positive displacement, and being below it means negative displacement. This tells you location relative to the origin, not necessarily the current direction of motion.
Straight line: A straight displacement-time graph means constant velocity because equal changes in time produce equal changes in displacement. This is the simplest type of motion to analyze because one gradient describes the whole interval.
Curved line: A curve means the velocity is changing, so you must think locally using tangents or compare gradients over intervals. This often signals acceleration or deceleration rather than uniform motion.
Check the graph type before doing anything: Examiners often test whether you can distinguish displacement-time graphs from distance-time or velocity-time graphs. This matters because the same visual feature, such as gradient, represents different physical quantities on different graph types.
Always inspect the starting value on the displacement axis: The object does not have to begin at the origin, so the initial displacement may be positive or negative. If you assume the graph starts at zero when it does not, every later interpretation of position and displacement change will be wrong.
Use units consistently: If displacement is in metres and time is in seconds, then gradients should be in metres per second. Careful unit tracking helps you catch errors, especially when a graph scale is uneven or when times and distances are read from different intervals.
Check the sign of the velocity: If the line slopes downward as time increases, the velocity must be negative. A positive answer in that situation usually means the subtraction order was wrong when computing gradient.
Check whether the answer fits the graph shape: A horizontal segment must correspond to zero velocity, and a steeper segment must correspond to a greater speed. These visual checks are fast ways to verify numerical calculations.
Distinguish total distance from displacement change: If the motion reverses direction, total distance travelled is larger than the magnitude of net displacement. This is a frequent source of lost marks when average speed and average velocity are both asked for.
Confusing negative displacement with negative distance: A negative displacement simply means the object is on the negative side of the origin according to the chosen sign convention. It does not mean a physically impossible negative distance has been travelled.
Thinking that crossing the axis means the object stops: Crossing the displacement axis only means the displacement is zero at that instant, so the object is at the origin. The object may still be moving through the origin if the gradient at that point is not zero.
Assuming a higher point means a greater speed: The height of the graph shows position, not velocity. Speed depends on the steepness of the graph, so a lower point on a steep line can represent faster motion than a higher point on a shallow line.
Using endpoint slope as instantaneous velocity on a curve: On a curved graph, the slope between two nearby but distinct points gives an average velocity over that interval, not the exact velocity at one instant. To estimate instantaneous velocity correctly, you need the tangent gradient at the chosen point.
Ignoring direction when calculating averages: If an object moves forward and then back, net displacement may be small even though total distance is large. This means average velocity and average speed can differ substantially, so you must decide which quantity the question asks for.