Derivation of Three Equations of Motion by Graphical Method

The equation of motion by graphical method explains how the three equations of motion are derived using a velocity-time graph. Instead of using algebraic formulas directly, this method uses the concepts of slope and area under a graph to obtain the equations of motion in a simple and visual way.

In this article, you will learn how to derive equations of motion by graphical method, including the derivation of the first, second, and third equations of motion in a step-by-step and easy-to-understand manner.

Table of Contents

• What Is an Equation of Motion
• Derivation of First Equation of Motion by Graphical Method
• Derivation of Second Equation of Motion by Graphical Method
• Derivation of Third Equation of Motion by Graphical Method

What Is an Equation of Motion?

An equation of motion is a mathematical relation between displacement, velocity, acceleration, and time for an object moving with uniform acceleration.

These equations help us calculate unknown quantities of motion without measuring each value directly.

The three equations of motion are:

• First Equation of Motion:  v=u+at
• Second Equation of Motion:  s=ut+12at2
• Third Equation of Motion:  v2=u2+2as

Where (u) is the initial velocity, v is the final velocity, a is the acceleration, t is the time taken, and s is the displacement. 

Now, there's an interesting question that comes into the picture. Why do we derive equations using graphs?

The fact is that a velocity-time graph gives two important pieces of information:

• The slope of the graph gives acceleration.

• The area under the graph gives displacement.

Using these two ideas, we can derive all three equations of motion in a simple and visual way.

Read More: Equation of Motion

Derivation of First Equation of Motion By Graphical Method

The derivation of first equation of motion can be done using different methods, such as the graphical method, algebraic method, and calculus method. Let's first see how this equation is obtained using a velocity-time graph.

We know that the slope of a velocity-time graph represents acceleration.

Acceleration=Slope of the graph

From the graph,

a=ABBC

According to the velocity-time graph,

AB=v−uandBC=t

Substituting these values, we get,

a=v−ut

Multiplying both sides by t,

at=v-u

Adding uuu to both sides,

v=u+at

Hence, the first equation of motion is,

v=u+at

This equation gives the relationship between initial velocity (u), final velocity (v), acceleration (a), and time (t) for an object moving with uniform acceleration.

Derivation of Second Equation of Motion by Graphical Method

The derivation of the second equation of motion can also be done using different methods. Here, we will derive it using a velocity-time graph. In this method, the displacement of the object is found from the area under the graph.

We know that the area under a velocity-time graph represents the displacement of the body.

Displacement(s)=AreaofOADC

The figure OADC consists of a rectangle OABC and a triangle ABD.

Therefore,

s=Area of Rectangle OABC+Area of Triangle ABD

s=(OA×OC)+12(AD×BD)

According to the graph,

OA=u,OC=t,AD=t,BD=v−u

Substituting these values, we get

s=(u×t)+12[t(v−u)]

From the first equation of motion,

v-u=at

Substituting v−u=at,

s=ut+12(t×at)

s=ut+12at2

Hence, the second equation of motion is,

s=ut+12at2

This equation gives the displacement of an object moving with uniform acceleration in terms of its initial velocity, acceleration, and time.

Derivation of Third Equation of Motion By Graphical Method

The derivation of third equation of motion can also be done using a velocity-time graph. In this method, we use the area under the graph and then remove the time term to obtain an equation involving only velocity, acceleration, and displacement.

We know that the area under a velocity-time graph gives the displacement of the body.

The area under the graph is the area of the trapezium OABC.

s=Area of trapezium OABC

The area of a trapezium is given by,

Area=12×(Sum of parallel sides)×Height

Therefore,

s=12(OA+CB)×OC

From the graph,

OA=u,CB=v,OC=t

Substituting these values, we get

s=12(u+v)t

From the first equation of motion,

v=u+at

Rearranging,

t=v−ua

Substituting the value of t in the above equation,

s=12(u+v)(v−ua))

s=(u+v)(v−u)2a

Using the identity,

(a+b)(a−b)=a2−b2

we get,

s=v2−u22a

Multiplying both sides by 2a,

2as=v2−u2

Adding  u2to both sides,

v2=u2+2as

Hence, the third equation of motion is,

v2=u2+2as

This equation directly relates the final velocity, initial velocity, acceleration, and displacement without involving time.

Read More: Derivation of Equations of Motion by Algebraic and Calculus Methods

The equation of motion by graphical method provides a simple and visual way to understand how the three equations of motion are obtained. By using the slope and area under a velocity-time graph, we can derive relationships between velocity, displacement, acceleration, and time without directly using algebraic formulas.