By using a graphical method Derive the kinematical equation of motion i v u at ii v2 u2 2as

In the given figure, the distance travelled (s) by a body in time (t) is given by the area of the figure OABC which is a trapezium.Distance travelld = Area of the trapezium OABCSo, Area of trapezium OABC,= `"(Sum of parallel sides)(Height)"/2`=`"(OA+CB)(OC)"/2`Now, (OA + CB) = u + v and (OC) = t.Putting these values in the above relation, we get:`s = ((u+v)/2)t` ....(1)Eliminate t from the above equation. This can be done by obtaining the value of t from the first equation of motion.v = u + atSo,`t = "v-u"/a`Now, put this value of t in equation (1), we get: `s = (((u+v)(v-u))/(2a))`On further simplification,

2as = v2 – u2

Finally the third equation of motion.`v^2 = u^2 + 2as`where(s) - Displacement(u) - Initial velocity(a) - Acceleration(v) - Final velocity

(t) - Time taken

Equations of motion of kinematics describe the basic concept of the motion of an object such as the position, velocity or acceleration of an object at various times. These three equations of motion govern the motion of an object in 1D, 2D and 3D. The derivation of the equations of motion is one of the most important topics in Physics. In this article, we will show you how to derive the first, second and third equation of motion by graphical method, algebraic method and calculus method.

Equations of motion, in physics, are defined as equations that describe the behaviour of a physical system in terms of its motion as a function of time.

There are three equations of motion that can be used to derive components such as displacement(s), velocity (initial and final), time(t) and acceleration(a). The following are the three equations of motion:

First Equation of Motion :
\(\begin{array}{l}v=u+at\end{array} \)
Second Equation of Motion :
\(\begin{array}{l}s=ut+\frac{1}{2}at^2\end{array} \)
Third Equation of Motion :
\(\begin{array}{l}v^2=u^2+2as\end{array} \)

To brush up on the basics of motion, refer to the article listed below::

The equations of motion can be derived using the following methods:

Derivation of equations of motion by Simple Algebraic Method
Derivation of equations of Motion by Graphical Method
Derivation of equations of Motion by Calculus Method

In the next few sections, the equations of motion are derived by all the three methods in a simple and easy to understand way.

For the derivation, let us consider a body moving in a straight line with uniform acceleration. Then, let the initial velocity be u, acceleration is denoted as a, the time period is denoted as t, velocity is denoted as v, and the distance travelled is denoted as s.

We know that the acceleration of the body is defined as the rate of change of velocity.

Mathematically, acceleration is represented as follows:

\(\begin{array}{l}a=\frac{v-u}{t}\end{array} \)

where v is the final velocity and u is the initial velocity.

Rearranging the above equation, we arrive at the first equation of motion as follows:

\(\begin{array}{l}v=u+at\end{array} \)

The first equation of motion can be derived using a velocity-time graph for a moving object with an initial velocity of u, final velocity v, and acceleration a.

In the above graph,

The velocity of the body changes from A to B in time t at a uniform rate.
BC is the final velocity and OC is the total time t.
A perpendicular is drawn from B to OC, a parallel line is drawn from A to D, and another perpendicular is drawn from B to OE (represented by dotted lines).