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# Graphical Representation of Motion

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#### Graphical Representation of Motion - Lesson Summary

A graph is a pictorial representation of the relation between two sets of data of which one set is of dependent variables and the other set is of independent variables.

In a displacement-time graph, displacement is a dependent quantity, taken on the Y-axis and time is taken on the X-axis, as it is independent. If the position of an object changes with respect to the reference position and time, it is said to be in motion.

The velocity at any instant of time is called instantaneous velocity. A positive slope of velocity-time graph gives acceleration and a negative slope gives deceleration or retardation of the object.

For an object moving with uniform acceleration, we have the following equations of motion,

v = u + at

s= ut +½ at2

v2 - u2 = 2as

Sn= u+a(n - ½)

Where u = Initial velocity,
s = Displacement,
a = Acceleration,
t  = Time and
v = Final velocity
Sn = The displacement of the body in n th second
n = n th second

If an object moves along a straight path, it is in a linear motion. If an object covers equal angular displacements in equal intervals of time then it is said to be in uniform circular motion. If an object repeats its motion on the path in regular intervals of time than it is in periodic motion.

Displacement-Time Graph
In the displacement-time graph, the time is taken on X-axis and the displacement of body is taken on Y-axis. From this graph, we can determine the velocity of body. Since, velocity is the ratio of displacement and time  therefore the slope of displacement-time graph gives the velocity. If the is positve, it implies that the body is moving away from the starting point.

If a body moves in a straight path the distance and the displacement of the motions are equal so the Displacement - Time graph and the Distance - Time graph are same.

(i) If the graph is parallel to time axis, then body is stationary.

(ii) If graph is a straight line, making an angle with the vertical axis i.e., Time axis  then body is moving with a uniform velocity. The velocity can be found out by finding the slope of the graph.

(iii) The graph can never be parallel to displacement axis, as it means that displacement increases indefinitely, with out any increase in time, which is impossible.

(iv) If graph is a curve, it means the body is moving with a variable velocity, and hence has some acceleration.

Velocity-Time Graph

In the velocity-time graph, time is taken on X-axis and the velocity is taken on Y-axis. Since velocity is a vector quantity, the positive velocity means that the body is moving in a certain direction away from its initial position and the negative velocity means that the body is moving in opposite direction.

(i) If Velocity-Time grapgh lies on the  X -  axis as the following

(a) it represents a body is at rest.
(b) Its acceleration is zero.

(ii) If Velocity-Time grapgh is parallel to time axis,

(a) Body is moving with uniform velocity.
(b) Its acceleration is zero.
(c) Its displacement can be found by finding the area of the graph.

(ii) If Velocity-Time grapgh is a straight line making an angle with time axis, as the following

(a) Body is moving with variable velocity.
(b) Its has uniform acceleration, which can be found by the slope of graph.
(c) Displacement can be found, by finding area under the velocity-time graph.
(d) If slope is positive, then the body has positive acceleration and vice-versa.
(e) If slope is negaitive, then the body has negative acceleration and vice-versa as shown in the following graph.

(iii) If the Velocity-Time grapgh is a curve,

(a) The body has variable velocity and variable acceleration.
(b) Area under the curve represents displacemet.
(c) Acceleration at aby instant can be found by finding slope at the point.

Acceleration-Time Graph
In the acceleration - time graph, time is taken on X-axis and acceleration is taken on Y-axis. From this graph, we can find the change in the speed in a certain interval of time. For linear motion , acceleration × time = change in speed, therefore the area enclosed between the acceleration - time sketch and the time axis gives the change in speed of the body.