Position-Time Graph: Object in Motion

Motion is the change within the position of a particle or an object with time. Here we tend to confine the study of the motion of objects on a straight line, referred to as linear motion. There are three main kinematics graphs: displacement-time graphs, velocity-time graphs, and acceleration-time graphs. Graphical illustration makes it simpler for us to understand the information. When analysing motion, graphs representing values of various parameters of motion make it easier to solve problems.

Position Time Graph

A graph drawn by taking time on the x-axis and displacement on the y-axis is termed a position-time graph. The principle is that the slope of the line on a position-time graph reveals useful data concerning the velocity of the object. The motion of an object is often represented by a position-time graph. Using this we can analyse completely different aspects of the motion of an object.

Position Time Graph of a Body at Rest

The position-time graph for a stationary object is a straight line parallel to the time axis as shown below. The displacement of a body at rest remains the same till it starts moving

Position Time Graph

Position-Time Graph for Uniform Motion

If an object is moving along a straight line and covers equal distances in equal intervalsof time, then it is said to be in uniform motion. Refer to the figure given below for such a motion.

Position-time Graph for Uniform Motion

How to Find Velocity from Position Time Graph?

When the object is in motion, its position changes with time. How fast the position is changing with time, is what we called as average velocity. That is average velocity can be defined as the change in position or displacement of an object with time known as average velocity (v).

\[v=\frac{{{x}_{2}}-{{x}_{1}}}{{{t}_{2}}-{{t}_{1}}}=\frac{\delta x}{\delta t}\]

Here, x1 and x2 are the positions of the object at times t1 and t2.

The SI unit of velocity is m/s. Velocity is also a vector quantity that possesses both magnitude and direction.

The average velocity can be positive or negative depending on the sign of the displacement we consider. If the displacement is zero then the velocity is also zero. The figure given below shows the x-t graphs under different conditions.

Different Position Time Graphs

The velocity can be found from the slope of position time graphs. For the first figure, the slope is zero means the velocity is zero. For the second one, the slope is positive and hence the velocity is positive. For the third graph, the slope of the graph is negative so is the velocity. One more point to note here is the steeper the slope the higher the value of velocity.

Position vs Time Graph and Relation with Velocity

Average velocity which was defined earlier only involves the displacement of the object. The magnitude of the displacement can be different from the actual path length, so we introduce another parameter known as average speed.

Average speed is defined as the total path length travelled divided by the time interval during which the motion took place.

$Average\text{ }spe\text{ed=}\frac{Total\text{ }distance}{Total\text{ }time}$

This quantity is always positive, unlike the average velocity. If the motion of the object is along the straight line and in the same direction then the velocity is positive. We can say that the magnitude of the average velocity is equal to the average speed.

Position Time Graph Under Uniform Acceleration

Acceleration

When the velocity of an object changes its course of motion with time, it is known as acceleration. Thus, we can say acceleration is the rate of change of velocity with time.

The average acceleration can be defined over a time interval as follows:

$a=\frac{{{v}_{2}}-{{v}_{1}}}{{{t}_{2}}-{{t}_{1}}}$

Here v₂ and v₁ are the final and initial velocities. The unit of acceleration is m/s². The acceleration at an instant is the slope of the tangent to the v-t graph. As we define velocity based on the slope we obtain, acceleration also depends on this, which means acceleration also can be positive, negative, or zero. Here we are constraining ourselves to the motion under uniform acceleration for the position-time graph.

For uniformly accelerated motion along the straight line, the position time graph is as follows.

Position Time Graph Under Uniform Acceleration

The relation between the x₀ the initial position, x, the final position, time t, and acceleration a with the initial velocity u0 is given below

$x={{x}_{0}}+{{u}_{0}}+\frac{1}{2}a{{t}^{2}}$

An object is said to be in uniformly accelerated motion when its velocity increases by an equal amount in equal intervals of time when travelling in a straight line.

Summary

A position-time graph is a graph that shows the changes within the values of a variable over time. Time is usually represented on the vertical axis, and therefore the values are on the horizontal axis of a position-time graph. When an object is moving in a uniform motion, the objects travel an equal distance in equal intervals of time. So, for the object moving in uniform motion, its position-time graph will be a straight line with relation to the time axis. We can use a position-time graph to help us understand the motion of an object.