Graphs of Motion – Introduction, Motion and Uniformly Accelerated Motion

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:

1. An equation between the variables defines the shape of the graph, such as a line, pair of straight lines, circle, etc.
2. If the graph is passing through the origin, then x=0 and also y=0. Suppose no graph is passing through the origin.
3. Differentiating y with respect to x can give the slope of the graph at that point.
4. To find out the area covered in the graph, integrate y along the x-axis.
5. 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=s₀ + vt

Important Points to remember:

1. The st graph is linear. Hence it is a straight line.
2. As v is constant, then the slope of the curve is constant.
3. As a=0, then the slope of the curve is zero.
4. The S=vt graph starts from the origin.
5. s=s₀ + vt graph may or may not start from the origin as displacement can be taken from any point on the graph.

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+1/2at² or s=1/2at²

s=s₀+ut+1/2at²  or s₌s₀+1/2at² 

Important Points to remember:

1. v-t is a straight line as it is linear in the graph.
2. s-t is a parabola, as all the s-t equations are quadratic in the graph.
3. 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.
4. 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-1/2at²

Important Motion:

1. The V-t graph cannot pass through the origin. Hence u≠0, the slope of the s-t graph is not zero.
2. Velocity keeps decreasing from positive to zero.
3. The slope of the v-t graph gives instantaneous acceleration, which is negative and decreasing.
4. 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:

1. If a =-9.8, then the motion is under gravitational force
2. In the graphs, O is the origin.
   v=u and slope tanϴ=u
3. In (graph 3) A is the maximum height of the ball reached
   v=0 and the slope tanϴ=0
4. B in (graph 3) is returning the ball to the ground
   v=-u and the slope tanϴ=-u