Derivation of Equations of Motion by Algebraic and Calculus Methods

Derivation of equation of motion helps us understand how the three important equations of motion are formed using the relationship between velocity, acceleration, displacement, and time. These equations are widely used to study the motion of cars, bicycles, falling objects, and many other moving bodies.

Have you ever wondered how scientists can calculate the distance travelled by a moving object or predict its final speed? The answer lies in the derivation of the equations of motion. This article explains the derivation of the first, second, and third equations of motion in a simple and easy-to-follow manner.

Table of Contents

• Understanding Equation of Motion
• Derivation of First Equation of Motion
• Derivation of Second Equation of Motion
• Derivation of Third Equation of Motion

Understanding Equation of Motion

An equation of motion is a mathematical relationship between displacement, velocity, acceleration, and time for an object moving with uniform acceleration. These equations help us calculate unknown quantities of motion without directly measuring them.

Interestingly! Only three equations are enough to solve most problems related to uniformly accelerated motion. 

The three equations of motion are,

First Equation of Motion:v = u + at Second Equation of Motion: s = ut + ½at² Third Equation of Motion: v² = u² + 2as

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

The equations of motion can be derived using different mathematical approaches. Depending on the level of study and the information available, physicists use various methods to obtain these equations. The three main methods of derivation are given below:

• Algebraic Method: Uses mathematical relationships between velocity, acceleration, displacement, and time.
• Calculus Method: Uses differentiation and integration to derive the equations.
• Graphical Method: Uses velocity-time graphs to derive the equations of motion.

Here we will mainly focus on simple derivations, and we will use the algebraic and calculus methods to derive the three equations of motion.

Read More: Equation of Motion

Derivation of First Equation of Motion

The first equation of motion gives the relationship between final velocity, initial velocity, acceleration, and time. Let's find out how this equation is derived using both algebraic and calculus methods. 

Algebraic Method

Acceleration is defined as the rate of change of velocity with respect to time. This is the starting point of all motion equations.

Now, we know that acceleration = change in velocity/time taken

So, a = (v − u) / t

Now, moving ahead, we rearrange the equation.

Multiply both sides by time:

at = v − u

Now, add u on both sides,

v = u + at

Calculus Method

By the definition of acceleration, it is the rate of change of velocity with respect to time.

a=dvdt

Multiplying both sides by dt, we get

adt = dv

Integrating both sides, where the limits of time are from 0 to t, and the limits of velocity are from u to v,

∫0tadt=∫uv

Since acceleration is constant,

a∫0tdt=∫uvdv

a[t]0t=[v]uv

a(t−0)=v−u

at=v−u

Adding u to both sides, we get

v=u+at

Thus, both the algebraic method and the calculus method give the same first equation of motion.

Derivation of Second Equation of Motion

The second equation of motion helps us calculate the displacement of an object moving with uniform acceleration. Now, there's an interesting question that comes into picture, how do displacement, velocity, and time combine to form this equation? 

Let's see the derivation of the second equation of motion step by step. 

Algebraic Method

The second equation of motion explains how displacement depends on time, initial velocity, and acceleration when an object moves with uniform acceleration.

Now the obvious question is, how do we get this formula?

We know that displacement is the shortest distance between the initial and final positions.

Now, if acceleration is constant, then displacement can be written as,

Displacement = average velocity × time

So, s = average velocity × time …(eq i)

Step 1: Finding average velocity

For uniform acceleration, the average velocity is:

Average velocity = (initial velocity + final velocity) / 2

So, vₐᵥ = (u + v) / 2

Now substitute this in equation (i)

s = (u + v) / 2 × t

Step 2: Use the first equation of motion

From the first equation of motion, we know, 

v = u + at

Now, substituting this value of v, we get

s = (u + u + at) / 2 × t

Step 3: Simplify the expression

s = (2u + at) / 2 × t

Now separate terms,

s = (2u × t) / 2 + (at × t) / 2

s = ut + ½at²

Calculus Method

As velocity is defined as the rate of change of displacement,

v=dsdt

Multiplying both sides by dtdtdt, we get

ds=vdt

From the first equation of motion,

v=u+at 

Substituting the value of v, we get

ds=(u+at)dt 

ds=udt+atdt 

Integrating both sides, where the limits of displacement are from 0 to sss and the limits of time are from 0 to t,

∫0sds=∫0tudt+∫0tatdt

Since u and a are constants,

⇒∫0sds=u∫0tdt+a∫0ttdt

⇒s−0=u(t−0)+a(t22−0)

⇒s=ut+12at2

Hence, the second equation of motion is

⇒s=ut+12at2

Derivation of Third Equation of Motion

The third equation of motion directly connects velocity, displacement, and acceleration without using time. This equation is especially useful when the time taken by the object is unknown. 

Moving ahead, let's see the derivation of the third equation of motion using two different methods. 

Algebraic Method

We know that displacement is equal to the product of average velocity and time taken.

So, s = Average velocity × Time

For uniform acceleration,

Average velocity = (u + v)/2

Therefore,

s = (u + v)/2 × t …(eq i)

Step 1: Find the value of time

From the first equation of motion,

v = u + at

After rearranging, we get

t = (v − u)/a

Step 2: Substitute the value of t in equation (i)

s = (u + v)/2 × (v − u)/a

Multiply both sides by 2a,

2as = (u + v)(v − u)

Using the identity,

(a + b)(a − b) = a² − b²

we get,

2as = v² − u²

Now add u² to both sides:

v² = u² + 2as

Calculus Method

As velocity is known as the rate of change of displacement,

v=dsdt...(i)

And we already know that acceleration is defined as the rate of change in velocity.

a=dvdt...(ii)

From equations (i) and (ii), we get

adsdt=vdvdt

⇒ads=vdv

Integrating both sides, where the limits of dsdsds are from 0 to s and the limits of dv are from uuu to v,

∫0sads=∫uvvdv

⇒a∫0sds=∫uvvdv

⇒a(s−0)=v2−u22

⇒2as=v2−u2

⇒v2=u2+2as

Hence, the third equation of motion is

v2=u2+2as

Read More: Equation of Motion by Graphical Method

Till now, we have learned that the three equations of motion can be derived step by step using simple mathematical methods. These equations make it easier to calculate the speed, displacement, and acceleration of moving objects and form the foundation of many concepts in mechanics.