Derivation of 3 Equations of Motion by Graphical Method

Introduction:
The equations of motion are a set of equations that describe the motion of an object. They are used to find the position, velocity, and acceleration of an object at any given time. The three equations of motion are derived using the graphical method.

Step 1: Graphical Representation
To derive the equations of motion, the motion of an object is first represented graphically. The position of the object is plotted against time to get a position-time graph. The slope of this graph gives the velocity of the object.

Step 2: Deriving the First Equation of Motion
The first equation of motion relates the displacement of an object with its initial velocity, final velocity, and time. This is given by:

S = ut + (1/2)at^2

where S is the displacement, u is the initial velocity, a is the acceleration, and t is the time taken.

To derive this equation, consider a position-time graph where the object starts from rest (u=0) and accelerates at a constant rate. The area under the graph gives the displacement of the object. The area is given by the area of the trapezium formed by the graph. This can be calculated as:

Area = (1/2)(t)(u + v)

where v is the final velocity of the object. Equating this to the displacement, we get:

S = (1/2)(t)(u + v)

Simplifying this equation, we get:

S = ut + (1/2)at^2

which is the first equation of motion.

Step 3: Deriving the Second Equation of Motion

The second equation of motion relates the final velocity of an object with its initial velocity, acceleration, and displacement. This is given by:

v^2 = u^2 + 2aS

where v is the final velocity, u is the initial velocity, a is the acceleration, and S is the displacement.

To derive this equation, consider a position-time graph where the object accelerates at a constant rate. The slope of the graph gives the velocity of the object. The final velocity, v, is given by the slope of the tangent to the graph at the final point. The initial velocity, u, is given by the slope of the tangent to the graph at the initial point. The displacement, S, is given by the area under the graph.

Using the formula for the area of a trapezium, we get:

S = (1/2)(u + v)t

Substituting this in the equation for final velocity, we get:

v^2 = u^2 + 2a[(1/2)(u + v)t]

Simplifying this equation, we get:

v^2 = u^2 + 2aS

which is the second equation of motion.

Step 4: Deriving the Third Equation of Motion

The third equation of motion relates the displacement of an object with its initial velocity, final velocity, and acceleration. This is given by:

S = (v + u)(t/2)

where S is the displacement, u is the initial velocity, v is the final velocity, and t is the time taken.

To derive this equation, consider a position-time graph where the object starts from rest (u=0) and accelerates at a constant rate. The