Derivation of Equation of Motion: Methods and Solved Examples

Derivation of the equation of motion is the mathematical method through which the three equations of motion are derived. Algebraic, Graphic, and Calculus methods are used to calculate various parameters such as time, velocity, acceleration, or distance in a given problem of motion related to daily life. Of these parameters, u, v, a, and s are vector quantities. Vectors going in one direction are positive and vectors in the opposite direction are negative. Equations of motion apply to uniformly accelerated motion. Moving objects have momentum, and forces cause variations in them. The total momentum in an explosion or collision is conserved and stays the same.

Read More: Potential Energy

Key Terms: Equations of Motion, Equations of Motion Derivation, Moment Inertia, Motion, Distance, Acceleration, Velocity, Time

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Equations of Motion

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There are three equations of motions in physics used to calculate various parameters such as time, acceleration, Velocity and Distance. These equations are as follows.

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

Where

• v is final Velocity 
• u is initial velocity 
• t is the time taken 
• a is acceleration
• s is the distance travelled

Equations of Motion

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Methods to Derive Equations of Motion

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There are three methods used to derive the equations of motion, these methods are as follows: 

1. Derivation using Algebraic Method
2. Derivation using Graphical Method
3. Derivation using Calculus Method

Motion in a Straight Line Video Explanation

Also Read:

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Derivation of First Equation of Motion

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Let’s imagine a moving body, travelling with uniform acceleration in a straight line. So, let the acceleration be a, initial velocity be u, the time period be t, distance travelled be s and final velocity be v.

Algebraic Method

The acceleration of a body is said to be the rate of change of velocity.

So, acceleration is represented mathematically as: a = (v-u)/t

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

By Rearranging the equation, we get the first equation of motion as: v = u + at

Graphical Method

In the below graph,

• The velocity changes in time t from A to B at a uniform rate.
• BC is the v and OC is t.
• A perpendicular is drawn from B to OE, A parallel line is drawn from A to D and A perpendicular is drawn from B to OC.

Derivation of First Equation of Motion

So, we know from the graph that

BC = BD + DC

Hence, v = BD + DC

and v = BD + OA (since DC = OA)

Therefore, v = BD + u (since OA = u) (Equation 1)

Now,

a = slope of line AB

a = BD/AD

Since AD = AC = t,

BD = at (Equation 2)

Equation 1 + equation 2, we get: v = u + at

Read More: Angular Momentum

Calculus Method

Since acceleration is the rate of change of velocity,

a=dv/dt

Rearranging the above equation, we get

adt=dv

Integrating both sides, we get

∫^t₀ adt = ∫^v_udv

at = v−u

Rearranging, we get v=u+at

Read More: Equations of Motion

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Derivation of Second Equation of Motion

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The second equation of motion can be derived via the following methods:

Algebraic Method

The rate of change of displacement is known as velocity.

Velocity = Displacement/Time

Displacement = Velocity×Time

Use average velocity in the place of velocity, if the velocity is not constant.:

Displacement={(Initial Velocity + Final Velocity)/2} × Time

s={(u+v)/2} × t

s=[{u+ (u + at)}/2] × t

s=[{2u + at/2}]×t

s=({2u/2} + {at/2})×t

s=(u + ½ at)×t

On further simplification, the equation becomes s=ut+½ at²

Read More: Application and Uses of Convex Mirror, Examples

Graphical Method

From the graph above, 

Distance travelled (s) = Area of figure OABC = Area of rectangle OADC + Area of triangle ABD

s=(½ AB×BD)+(OA×OC)

Derivation of Second Equation of Motion

Since BD = EA,

s=(½ AB×EA) + (u × t)

As EA = at, 

s=½ × at × t+ ut

So, the equation becomes s= ut+ ½ at²

Calculus Method

The rate of change of displacement is known as velocity.

Mathematically, this is:

v=ds/dt

ds=vdt

ds=(u + at) dt

ds=(u + at) dt = (udt + atdt)

On further simplification, the equation becomes:

∫₀^sds=∫₀^tudt+∫₀^tatdt

s = ut + ½ at²

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Derivation of Third Equation of Motion

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The third equation of motion can be derived by: 

Algebraic Method

Displacement is an object’s rate of change of position. Mathematically, this is:

Displacement=(Initial Velocity+Final Velocity/2) × t

s=(u+v/2) × t

From the first equation of motion, we know that

v= u + at

t = (v−u)/a

s=(v+u/2) (v−u/a)

s=(v²−u²/2a)

2as=v²−u²

That results in v² = u² + 2as

Read More: Types of Motors and its Application

Graphical Method

s is the total distance travelled given by the Area of trapezium OABC.

S = Height × ½ (Sum of Parallel Sides)

S=(OA+CB)×OC

Derivation of Third Equation of Motion

Here, CB = v, OA = u and OC = t, 

so,

s= ½ (u+v) × t

t = (v – u)/ a

s= ½ ((u+v) × (v-u))/a

s= ½ (v+u) × (v-u)/a

s = (v²-u²)/2a

Third equation of motion is: v² = u² + 2as

Read More: Unit of Frequency

Calculus Method

The rate of change of velocity is Acceleration:

a= dv/dt (1)

The rate of change of displacement is velocity.

v= ds/dt (2)

Cross multiplying (1) and (2), we get

ads/dt=vdv/dt

∫₀^sads=∫_u^vvds

as = (v²−u²)/2

This results in v² = u² + 2as

Read More: Capacitor Types

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Things to Remember

• Derivation of the equation of motion is the mathematical method through which the three equations of motion are derived using different methods, these equations are used to calculate various parameters such as time, velocity, acceleration or distance in a given problem of motion related to daily life. 
• When you know the derivation of a given equation, you tend to understand the basic concept behind that equation, and that in turn helps to solve problems in a better way.
• let the acceleration be a, initial velocity be u, the time period be t, distance traveled be s and final velocity be v.
• There are three methods used to derive the equations of motion, these methods are as follows: Derivation using Algebraic Method, Derivation using Graphical Method, Derivation using Calculus Method.

Read More:

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