 # 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.

• 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

1. Write different types of kinetic energies.

There are 5 types of kinetic energies:

• 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.