**Graphs of Motion**

**Explanation:**

**Introduction to the graphs:**

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

Let’s say *x* is an independent variable and *y* is the dependent variable. 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 C

- 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.

- 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:**

*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.

*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 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. 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 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. 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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