How to find displacement on a position time graph

Quick summary: Displacement on a position-time graph is found by taking the difference between the object’s final position and its initial position: $\Delta x = x_{final} – x_{initial}$. Read each position by projecting the point on the graph onto the y-axis. A positive result means displacement in the positive direction, a negative result means displacement in the negative direction, and zero means the object has returned to its starting position.

In this article, we will learn how to find displacement on a position-time graph. A position-time graph is a graph where the instantaneous position $x$ of a particle is plotted on the y-axis and the time $t$ is plotted on the x-axis. This graph is one of the key tools used to study motion and its types in physics, since it shows exactly where an object is located at any given instant.

Key takeaways:

• Displacement can be found on a position-time graph by drawing a line to get the projection of a point on the y-axis. This projection gives the position of the object at that instant.
• The displacement between two points on the graph is found by subtracting the initial position from the final position. A negative displacement means the object has moved in the negative direction.

Displacement of an object is defined as the change in its position in a fixed direction. It is given by the vector drawn from the initial position to the final position of the object.
Displacement is a vector quantity. So, it has both magnitude and direction. The displacement of an object can be positive, negative, or zero.

How to find displacement on a position-time graph

Consider the figure given below, which shows the position-time graph of an object.

From the figure above, the object starts its journey from the origin at initial time $t=0$ and position $x=0$. It reaches its final position at time $t=9\,s$, represented by point $H$ on the graph.

Let us now find the position of the object at point $A$ marked on the graph. We draw a line to get the projection of point $A$ onto the y-axis. This projection gives the position of the object at that instant. The figure below shows the projection of various points onto both axes.

The projection of point $A$ onto the y-axis is at $4\,m$ from the origin. So:

• Position of the object with respect to the origin at point $A$ = $4\,m$
• Position of the object with respect to the origin at point $B$ = $4\,m$

Since both $A$ and $B$ project to the same position on the y-axis, the object has not changed position between these two points, even though time has passed.

IMPORTANT NOTE:

• A flat, unchanging position on the graph means the object is at rest during that interval.
• Zero change in position over an interval means zero displacement, even if the object was in motion at other points during the journey.

Similarly, you can find the position of the object at any point with respect to the origin $O$.

Example 1: Displacement between $t=0\,s$ and $t=3.5\,s$

At $t=3.5\,s$, the object is at point $P$.

We know displacement is the change in position of the object. Here:

$x_0 = 0\,m$
$x_{3.5} = -4\,m$

So, change in position = displacement:

$$x_{3.5} – x_0 = -4 – 0 = -4\,m$$

Here, the displacement is negative. Negative displacement means the object has moved in the negative direction. So, instead of being 4 m forward from the origin, the object is 4 m backward from the origin. See distance and displacement for more on how the sign of displacement is interpreted.

Example 2: Displacement between $t=0\,s$ and $t=4\,s$

At $t=4\,s$, the object is at position $x=0\,m$.

Here:

$x_0 = 0\,m$
$x_4 = 0\,m$

So, change in position = displacement:

$$x_4 – x_0 = 0\,m$$

Thus, the displacement is zero. This means the object has returned to its starting position by $t=4\,s$, even though it may have moved during the interval.

Similarly, you can find the displacement between any two instants using the position-time graph.

Related articles

• How to find displacement on a velocity-time graph
• How to find average velocity on a position-time graph
• How to find instantaneous velocity on a position-time graph

Frequently Asked Questions

How do you find displacement from a position-time graph?

Project the object’s position at the starting time and the ending time onto the y-axis. Subtract the initial position from the final position: $\Delta x = x_{final} – x_{initial}$. This vertical difference is the displacement.

Is displacement the same as the length of the curve on a position-time graph?

No. Displacement is only the straight vertical difference between the initial and final positions on the y-axis. It is not the length of the path traced by the curve between the two points.

What does a negative displacement mean on a position-time graph?

A negative displacement means the object has moved in the negative direction from its starting point, not that it has moved backward in time or lost distance.

What does zero displacement mean if the object was moving?

Zero displacement means the object’s final position is the same as its initial position. The object may still have moved during the interval, but it ended up back where it started.

Can two different points on a position-time graph have the same position value?

Yes. If two points project to the same value on the y-axis, the object was at rest between them, since its position did not change even though time passed.