Velocity-Time and Position-Time Graphs

Displacement

Displacement is the difference in the initial and the final position of an object with respect to its origin. It is a vector physical quantity. The net displacement can be zero but distance can never be zero. The S.I. unit is a metre.

Distance

Distance is the actual path that is covered by the particle moving from one point to another. It is a scalar quantity. It can never be zero. 

Example

In the figure, if a particle is moving from A to E then we write the distance travelled by the particle along the curved path that is along ABCDE. If we talk about the displacement then it must be the straight-line joining point A and E. It is not always possible for displacement to be equal to the distance. It depends on the path covered by any object.         

Speed

Speed is the rate of change of position with respect to time. It is a scalar quantity which means it is independent of direction. We measure speed in the unit of m/sec.

Mathematically, speed is expressed as:

                                    S =  dx/dt.

Where:   s is the speed

               x is the position of the particle.

               It is the time taken by the particle while changing its position.

Velocity

In mechanics, we define velocity as the rate of change in displacement with respect to time. It is a vector physical quantity generally denoted by putting an arrow over v or sometimes bold letter v.

Mathematically, velocity is expressed as: 

                                           V = dX/dt

Where-  v is the velocity

               X is the displacement

             d/dt is the differentiation

The Slope

The slope of a curve at any point is defined as the tangent (tanθ) of the angle made by the tangent at the given point with the x-axis. It is denoted by m. If we take two points, such as (x1, y1) and (x2, y2), on the curve then the slope is written as,

                                               m(slope)= (y2-y1)/ (x2-x1) 

when we have to find the slope of a graph at a single point then we differentiate the function and find its value at that given point.

The slope of an x-t time graph leads us to the velocity and the slope of a velocity-time graph leads us to the acceleration.

In this figure, we can see if we are finding the slope of a position-time graph then we can write 

              Slope = dx/dt which is the velocity of the particle.

And if we do the same for the velocity-time graph we shall be able to define the acceleration.

Types of X-T Graphs 

On the X-axis we have time and on the Y-axis we have the displacement.

1. The position of the particle is fixed with respect to time. Means the body is at rest. 

Slope = dx/dt =0 

So, the particle has no velocity.

For example, when we are standing at a place then our position will not change with respect to the time that our body will be at rest. So, if we plot a graph between position and time, it will be a straight line parallel to the time axis. We have considered the x-axis to be the time axis. The graphs are as follows:

2. The slope of the straight line is a fixed (constant) value. So, the particle is in uniform motion and the velocity can be determined as follows: 

         V =slope =  dx/dt = constant

A negative slope indicates the particle is moving in the negative x-axis direction.

For example, suppose we are running at a constant speed of 2m/sec then it means we are covering a distance of 2m in each one sec. So, our position is changing linearly that is why we see a graph of straight lines having a fixed slope of 2.

3. In this graph, at x = 0, the slope and velocity are zero and at x = 4, the slope is positive. Therefore, the velocity is changing.

For example, if we throw a ball up in the sky then its position will change non linearly.

The ball reaches the highest point in the portion of the graph after the origin in the first half of the throw.

Again at this point, we can classify the motion of the particle into the following two categories:

1. Uniformly accelerated motion (dv/dt is fixed)

2. Non-uniformly accelerated motion (dv/dt is variable)                 

Types of V-T Graphs

On the X-axis we have time and on the Y-axis we have the velocity.

1. This velocity is fixed which means the particle is under uniform motion.

dv/dt is zero means no acceleration.

2. In this case, velocity is increasing.

Slope = dv/dt = constant

In the case of a negative slope, we get a decrease in velocity with time. Like someone pushing the particle in the negative x-direction.

3. In this we have slope = dv/dt = variable

Conclusion

Motion can be classified based on different types of graphs. When we trace the position of any object against time then it is called its position-time graph and when we trace the velocity against the time it is called a v-t graph. Position-time graphs and velocity-time graphs are very helpful to find out the other physical quantities such as acceleration. We can find other quantities if one of them is known. The constant x-t graph represents the body at rest and the constant v-t graph represents the body having a constant velocity. A V-t graph having a positive slope represents the body having acceleration in the x-direction and a negative slope represents retardation. Distance-time graphs can not be in the 3rd or 4th quadrant same as the speed-time graph. Distance and displacement may or may not be equal. If we get a position-time graph indicating two positions at a single time we discard them, that is a practically impossible situation.