Derivation of Work-Energy Theorem

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,

v² = u² + 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:

v² – u² = 2as

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

v² – u² = 2*a*d

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

½ mv² – ½ mu² = ma*d

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

Now, the above equation can be written as:

½ mv² – ½ mu² = 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, 

x₀= initial position

x₁ = Final Position

We know that, 

Kinetic energy KE= ½ *mv^2.

This means that when x₀ then K₀ and when x₁ then K₁.

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. v₁ to v₂ by applying acceleration, and the object gets displaced by a distance d.

Thus the equation becomes:

v₂²−v₁²= 2*a*d

We can also write it as:

a = (v₂²−v₁²)/2d, or

Now substituting the value of a in F= ma

F = m (v₂²−v₁²)/2d, or

Fd = m (v₂²−v₁²)/2d, or

Fd = ½ m*v₂² – ½ m*v₁² —Equation 1

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

In equation 1, the quantity

K₂ = m*v₂²/2 is the final Kinetic energy of the object.

K₁=m*v₁²/2 is the initial Kinetic of the object.

Thus equation (1) becomes

W=K₂-K₁=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.

Relevant Blogs

Who is Known as the Father of Modern Politics?	GK Quiz for Class 5
What is the Total Weight of Chandrayaan-3?	World Science Day for Peace and Development
National Science Day	National Techies Day

FAQs

For more information about such informative articles, make sure to check the trending events page of Leverage Edu.