Derivation of Kinetic Energy Formula

What is Kinetic Energy?

The word Kinetic energy came from the French word travail mécanique (mechanical work) or quantité de travail (quantity of work). The kinetic energy of a body is the energy possessed by the body by virtue of its motion.


For example, There are various applications of kinetic energy:

  • A nail is driven into a wooden block on the amount of kinetic energy of the hammer striking the nail.

  • A baseball thrown by a pitcher has a small mass but a large amount of kinetic energy due to high velocity.

On this page, we will learn about the following:

  • The formula for kinetic energy

  • Derive the formula of kinetic energy

  • Kinetic energy equation derivation

  • Kinetic energy derivation calculus

  • Derivation of kinetic energy using algebra


The Formula for Kinetic Energy

A dancing man is said to be more energetic than a snoring man. In physics, a moving particle is said to have more energy than the particle at rest. Quantitatively the energy of the moving particle is defined by:


                                                         →  → 

K (v)  = 1/ 2 x m x v ^ 2  = 1/ 2 m   v .  v


                                                

This is the kinetic energy of the particle.


Therefore, the sum of the kinetic energy of a system of particles is the sum of all its constituent particles. Which is given by:



 K =  ∑ 1/ 2 mi x vi ^ 2

         i


The kinetic energy of a particle or a system of particles can increase or decrease or remain constant as time passes.

Deriving The Kinetic Energy Formula by Algebra

The kinetic energy can be obtained by either of the following:

  • The amount of work done in stopping any moving object.

  • The amount of work done in giving the velocity to the body from the state of rest.

Let’s consider the second case:

Suppose m =  mass of the body at rest.

u = Initial velocity of the body.

                F = The force applied to the body

                a = acceleration produced in the body in the direction of force applied

               v = The velocity acquired by the body while moving a distance s.


K∆S = W = m∆a….(1)


Using the third equation of motion:


v ^ 2 -  u  ^ 2 = 2as…


a  = v ^2 - u ^2  / 2s…(2)


∵     F = ma…(3)


Putting the value of ‘a’ in eq(3)


F = m x (v ^2 - u ^ 2) / 2s….(4)


Combining eq(1) and (4), we get that:


∆K = m ( v ^2 - u ^2) / 2….(5)


Putting u = 0  in eq(5), we get that:


∆K = m ( v ^2 /2)......(6)


Work done on the body is given by:


W = Force x distance….(7)


This statement states that a work W is done by a body to move from one position to another by a distance s when the force F is applied to a body at rest.


Putting u = 0  in eq(2), we get that:


a  = v^2 / 2s

F = m  a

Therefore, F = m (  v ^2 /2s)


Putting the value of F in eq(7), we get that


W =  m x v ^2 / 2s x s 

W =  1/ 2 x m x v ^2


K.E of body = W = 1/ 2 x m x v ^ 2


This work done on the body is because of Kinetic energy (K.E) of the body.

Derivation of Kinetic Energy Formula by Calculus

The formula for kinetic energy can be obtained by the method of calculus:


Suppose

m = mass of a body

u =  Initial velocity of the body 

F  =  The force applied to the body in the direction of the motion

ds  = A small displacement of the body in the direction of motion


The small amount of work done by the force will be:

           →   →

dW =  F  . ds = F ds Cos 0°  = Fds  (∵  Cos 0° = 1)

                           →             →        →     →

dW = F ds = m  a ds    (∵  F = m   a  )  ( a  is an acceleration produced by the force)

               →                             →   → 

    =  m (dv  /  dt ) ds       (∵  a = dv / dt)


    =  m (ds /dt) dv


dW    = m v dv                   ( ∵  v = ds /dt)


Total work done by the force is increasing the velocity of the body from  u  to v is:

         v                                                    v                      v

W =   ∫    m v dv   = m [ v ^ (1+1) / (1+1) ] = m [ v ^2 / 2] 

          u                                                    u                      u


=> 1/ 2 m ( v^2 - u ^2)


W = 1 / 2 m  v^ 2 - 1/ 2 m u ^2…..(7)

Work energy theorem



     Work done = Final K.E. - Initial K.E.



W = change in kinetic energy


The work done by a force is a measure of the change in kinetic energy of the body which proves the work-energy principle.


On putting u = 0 in eq (7), we get that:


W = 1/ 2 m v^ 2 - 0 


K.E. of the body = W = 1/ 2 m v ^2

Summary

  • The force does some work on a body, the kinetic energy increases by the same amount. Thus, according to this principle, work and energy are equivalent to each other.


  • A force is necessary to change the kinetic energy of the particle. If the resultant force acting on a particle is perpendicular to its velocity, the speed does not change and hence the kinetic energy does not change.

SAQs

Q1: Calculate the average frictional force needed to stop a car weighing 600 kg at a distance of 35 m, if the initial speed is 36 km/h.

Solution:  Given m = 600 kg, s =35 m

                       v =  final velocity = 0  (Since the body will stop finally)

                        u= 36 km/h       

                u = 36 kmph = 10 m/s

According to work-energy principle,


W = Change in K.E. = 1/ 2 m (v^2 - u ^2)


F x s = 1/ 2 m (v^2 - u ^2)    


and, W = F x s 

F x 35 = 1 /2 x 600 x ( 0 - 10^2)

On solving, we get that:


Average frictional force (F) = 857 N



FAQ (Frequently Asked Questions)

1. Write different types of kinetic energies.

There are 5 types of kinetic energies:

1. Radiant energy

  • A type of energy created from electromagnetic waves such as X-rays, sunshine.

2. Thermal energy

  • Geothermal energy, geysers, boiling water

3.Electrical energy

  • Lightning, AC/DC devices, batteries use.

4. Mechanical energy

  • A bullet fired from a gun, wind energy.

5. Sound energy

  • Human voice, stereo speakers, a buzzing bee.

2. What does kinetic energy depend upon?

The kinetic energy of a body is half the product of the mass of the body and square of the velocity of the body.

                    

K.E. =   1/ 2 m v ^2


    We observed that the 

 K.E ∝ m and also 

K.E ∝ v ^ 2.

    A heavier body and a body moving faster possess greater kinetic energy and 

 vice-versa.