# What is the relationship between displacement velocity and acceleration graphs?

Displacement, s

Displacement is the difference between the initial position and the final position of an object.

Velocity, v

Velocity is the rate of change of displacement. Thus, $$v = {ds \over dt}$$

Acceleration, a

Acceleration is the rate of change of velocity. Thus, $$a = {dv \over dt} = {d^2 s \over dt^2}$$

A particle moves in a straight line such that, t seconds after passing a fixed point O, its velocity, v m/s, is given by v = t2 + 1.

Find

(i) an expression, in terms of t, for the acceleration of the particle,

(ii) and an expression, in terms of t, for the displacement of the particle.

(i) \begin{align} a & = {dv \over dt} \\ a & = {d \over dt} (t^2 + 1) \\ a & = 2t \end{align} (ii) \begin{align} s & = \int v \phantom{.} dt \\ s & = \int t^2 + 1 \phantom{.} dt \\ s & = {t^3 \over 3} + t + C \end{align}

The particle starts at fixed point O.

\begin{align} \text{When } & t = 0 \text{ and } s = 0, \\ 0 & = {(0)^3 \over 3} + (0) + C \\ 0 & = C \\ \\ \therefore s & = {t^3 \over 3} + t \end{align}

Understanding the numericals of Physics with the help of equations and derivations can be boring, but with the help of graphs, it becomes interesting as well as easy to understand what the solution is explaining. Graphs in Physics play a vital role as most of the concepts use them. In this article, let us know more about different graphs in detail.

A graph is defined as a pictorial representation of information which is a two-dimensional drawing explaining the relationship between dependent and independent variables. Independent variables are represented on the horizontal line known as the x-axis, while the dependent variables are represented on the vertical line known as the y-axis.

We have already seen the mathematical approach toward speed, velocity, distance and displacement. But graphs actually give us a better understanding of the motion. From the point of view of physics, one should be able to interpret motion by looking at graphs. Here we will be talking mainly about velocity time graph and displacement time graph.

The displacement of an object is defined as how far the object is from its initial point. In the displacement time graph, displacement is the dependent variable and is represented on the y-axis, while time is the independent variable and is represented on the x-axis. Displacement time graphs are also known as position-time graphs. There are three different plots for the displacement time graph, and they are given below:

• The First graph explains that the object is stationary for a period of time such that the slope is zero, which means that the velocity of the object is zero.
• the Second graph explains the velocity of the object, and hence the slope of the graph remains constant and positive.
• Third graph explains that the acceleration is constant. The slope of the graph increases with time.

The slope for the displacement time graph is given in the table below:

 $$\begin{array}{l}\frac{\Delta d}{\Delta t}\end{array}$$

Therefore, the following are the takeaway from the displacement time graph:

• Slope is equal to velocity.
• Constant velocity is explained by the straight line, while acceleration is explained by the curved lines.
• Positive slope means the motion is in the positive direction.
• Negative slope means the motion is in the negative direction.
• Zero slope means that the object is at rest.

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In the velocity-time graph, velocity is the dependent variable and is represented on the y-axis, and time is the independent variable, represented on the x-axis. The slope of the velocity time graph is given as in the table:

 $$\begin{array}{l}\frac{\Delta v}{\Delta t}\end{array}$$

We see that the slope of the velocity-time graph is the definition of acceleration; therefore, it can be said that the slope is equal to acceleration. Therefore, the following are the points understood from the slope:

• Steep slope represents the rapid change in velocity.
• Shallow slope represents the slow change in velocity.
• If the slope is negative, then the acceleration will also be negative.
• If the slope is positive, then the acceleration will also be positive.
• The area under the velocity represents the displacement of the object.

In the acceleration time graph, acceleration is the dependent variable and is represented on the y-axis, and time is the independent variable and is represented on the x-axis. The slope of the acceleration time graph is as given in the table:

 $$\begin{array}{l}\frac{\Delta a}{\Delta t}\end{array}$$

The slope of the acceleration time graph is known as jerk. The following are the points understood from the graph:

• If the slope is zero, then the motion is said to have constant acceleration.
• The area under the graph represents the change in velocity.

A graph is defined as a pictorial representation of information which is a two-dimensional drawing showing the relationship between dependent and independent variables. Independent variables are denoted on the horizontal line known as the x-axis, while the dependent variables are denoted on the vertical line known as the y-axis.

The main components of a 2D graph are the x-coordinate and the y-coordinate.

Displacement-time graph, velocity-time graph, and acceleration-time graph are three common types of graphs in classical mechanics.

In the displacement time graph, displacement is the dependent variable and is represented on the y-axis, while time is the independent variable and is represented on the x-axis and is also known as the position-time graph.

In the velocity-time graph, velocity is the dependent variable and is represented on the y-axis, and time is the independent variable, represented on the x-axis.

In the acceleration time graph, acceleration is the dependent variable and is represented by the y-axis, and time is the independent variable and is represented by the x-axis.

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