We do various activities in our daily life, sitting and reading a newspaper is also doing some work. According to **Physics**, any work done that does not involve the displacement of the body is not considered work. Thus, to describe the phenomenon of work energy we will derive the work energy theorem. The **work-energy** theorem explains the reasons behind the physics of no work. Stay tuned and continue reading this article to get the derivation of the work-energy theorem!

**Also Read: ****Class 11 Physics Chapter 3 – Motion in a Straight Line**

## What is Work?

Work is defined as the product of force and displacement. It means when the force acting on an object displaces the object from its original position, then it is said to be work done.

You must also know that moving objects possess **kinetic energy**. Thus, the work-energy theorem gives the relationship between work and kinetic energy.

Work is the change in kinetic energy.

**W= ΔK**

**Also Read: ****Class 9 Motion **

## Work-Energy Theorem

The work-energy theorem is defined as the work done by the net force applied on an object or body is directly proportional to the change in kinetic energy.

The formula of the Work-Energy Theorem is

**Kf – Ki = W**

Here,

Kf= Final kinetic energy

Ki= Initial Kinetic energy

Kf – Ki is a change in Kinetic energy

W= Work done on an object

## Derivation of Work-Energy Theorem

Equations of motion,

v^{2} = u^{2} + 2as

Here,

v = final velocity of an object

u = initial velocity of an object

a = constant acceleration

s = displacement of the object

The above equation can also be written as:

v^{2} – u^{2 }= 2as

Substituting the values of the vector quantities, the equation would be:

v^{2} – u^{2 }= 2*a*d

Now, multiply both sides of the equation by m/2, we get:

½ mv^{2 }– ½ mu^{2} = ma*d

According to Newton’s second **law of motion**, F= ma,

Now, the above equation can be written as:

½ mv^{2 }– ½ mu^{2} = F*d

We also know that W= F.d and, Kinetic energy = (mv²)/2,

Thus the equation becomes:

**K _{f }– K_{i }= W**

Hence, we have:

ΔK = W

Here ΔK = K_{f }– K_{i}

ΔK= Change in kinetic energy

**Also Read: ****Basic Physics Formulas and Notes for Competitive Exams **

## Derivation of Work-Energy Theorem for Variable Force

Here is the proof of the work-energy theorem for variable force:

F= Variable Force

t= Time

Work Done by Force is given by:

W= ∫^{X1}_{X0 }F* dx —equation 1

Here,

x_{0}= initial position

x_{1 }= Final Position

We know that,

Kinetic energy KE= ½ *mv^{2.}

This means that when x_{0 }then K_{0 }and when x_{1 }then K_{1}.

Now, differentiate K w.r.t time (t):

dK/dt= m*v* dv/dt

dK/dt=m*a*dx/dt

dK/dt= F*dx/dt

dK= F*dx

**∫ _{K0}^{K1} dK = ∫_{x0}^{x1} F* dx**

ΔK = W

Hence Proved.

**Also Read: ****Laws of Motion Class 11**

## Derivation of Work-Energy Theorem for Constant Force

From Newton’s Second Law of Motion: F = ma

Here,

a = acceleration of the object

The velocity of the object increases i.e. v_{1} to v_{2} by applying acceleration, and the object gets displaced by a distance d.

Thus the equation becomes:

v_{2}^{2}−v_{1}^{2}= 2*a*d

We can also write it as:

a = (v_{2}^{2}−v_{1}^{2})/2d, or

Now substituting the value of a in F= ma

F = m (v_{2}^{2}−v_{1}^{2})/2d, or

Fd = m (v_{2}^{2}−v_{1}^{2})/2d, or

Fd = ½ m*v_{2}^{2} – ½ m*v_{1}^{2} —Equation 1

Fd is the work done by the force F to move the object through a distance d.

In equation 1, the quantity

K_{2 }= m*v_{2}^{2}/2 is the final Kinetic energy of the object.

K_{1}=m*v_{1}^{2}/2 is the initial Kinetic of the object.

Thus equation (1) becomes

**W=K _{2}-K_{1}=ΔK- Equation 2**

Here,

ΔK = change in Kinetic energy of the object.

From equation 2, it is clear that the work done by a force is equal to the change in kinetic energy.

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## FAQs

**What is the formula of the Work-Energy Theorem?**

The Formula for the Work-Energy Theorem is W= Kf – Ki

Here, Kf= Final kinetic energy, Ki= Initial Kinetic energy

Kf – Ki is a change in Kinetic energy

W= Work done on an object

**What is the statement of the Work-Energy Theorem?**

The work-energy theorem states that the total work done on an object by the application of force is equal to the change in its kinetic energy.

**What is the derivation of the work equation?**

The derivation of the work equation is W= F*d. Here W= work and it is the product of Force (F) and Displacement (d).

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