## Graphs of Motion

### Introduction to the Graph

Generally, graphs are plotted between two variables, one is the independent variable, and another is the dependent variable.

Suppose *x* is an independent variable and *y* is the dependent variable. Then, if a graph is plotted between *x* and *y* (*x,y*), the following points should be remembered before plotting a graph.

### Points to be remembered:

- An equation between the variables defines the shape of the graph, such as a line, pair of straight lines, circle, etc.
- If the graph is passing through the origin, then
*x*=0 and also*y*=0. Suppose no graph is passing through the origin. - Differentiating
*y*with respect to*x*can give the slope of the graph at that point. - To find out the area covered in the graph, integrate
*y*along the*x*-axis. - The most asked graphs in motion are
*(s-t,v-t,a-t,*and*v-s*).

## Graphs of Uniform Motion

The equations which appear in uniform motion graphs are

*a*=0,*v *= const, *s=vt* or *s=s _{0} + vt*

### Important Points to remember:

*The st*graph is linear. Hence it is a straight line.- As
*v*is constant, then the slope of the curve is constant. - As
*a*=0, then the slope of the curve is zero. *The S=vt*graph starts from the origin.*s=s*graph may or may not start from the origin as displacement can be taken from any point on the graph._{0}+ vt

## Graphs of Uniformly Accelerated Motion

Generally, we encounter the following equations in graphs of uniformly accelerated motion:

*a* = 0(positive)

*v=u+at* or *v=at*

*s=ut+1/2at ^{2}* or

*s=1/2at*

^{2}*s=s _{0}+ut+1/2at^{2}* or

*s*

_{=}s_{0}+1/2at^{2 }### Important Points to remember:

*v-t*is a straight line as it is linear in the graph.*s-t*is a parabola, as all the*s-t*equations are quadratic in the graph.- The slope of the
*s-t*graph gives instantaneous velocity. Therefore, as instantaneous velocity is positive and constantly increasing, the graph*s-t*is also positive and constantly increasing. - The slope of the
*v-t*graph gives instantaneous acceleration, which is positive and increasing.

## Graphs of Uniformly Retarded Motion

We encounter the following equations using graphs of uniformly retarded motion

*a*=constant(negative)

*v=u-at*

*s=ut-1/2at ^{2}*

### Important Motion:

- The
*V-t*graph cannot pass through the origin. Hence*u*≠0, the slope of the*s-t*graph is not zero. - Velocity keeps decreasing from positive to zero.
- The slope of the
*v-t*graph gives instantaneous acceleration, which is negative and decreasing. - The slope of the
*s-t*graph gives instantaneous velocity. Therefore, as instantaneous velocity is positive and constantly decreasing, the graph*s-t*is also positive and constantly decreasing.

### Graphs of Uniformly Retarded Motion and then Accelerated Motion

In this case, the graphs are drawn, and the following points are noted for the use of graph

Let us consider a ball is thrown vertically upwards

### Important Points:

- If
*a*=-9.8, then the motion is under gravitational force - In the graphs,
*O*is the origin.

*v=u*and slope tanϴ=*u* - In (graph 3) A is the maximum height of the ball reached

*v*=0 and the slope tanϴ=0 *B*in (graph 3) is returning the ball to the ground

*v=-u*and the slope tanϴ=-*u*

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