# Work Energy Theorem for Constant and Variable Force

The Post will give you knowledge about all the concepts of the **Work-Energy theorem**. Welcome to TheCScience, in this article you will learn about the **Work-Energy Theorem and its Derivation**. The Post gives you information in brief so you can easily revise for your examination.

First, we will talk about what is a work-energy theorem. Then we will derive the formula for constant and variable force. So let’s talk about **what work-energy theory is.**

## Work Energy Theorem

According to this theorem, Work done is equal to a change in Kinetic energy.**W = Kf – Ki**

Now, we have to prove this theorem by derivation so firstly we will solve it for constant force and then solve it for variable force.**Note:-** Variable Force is Important from an examination point of view. We have to solve the variable force by integration method. So practice the derivations.

## Work Energy Theorem for Constant Force

**Work = Force x Displacement**

W = FxS / F = ma

W = mas**From 3rd equation of motion**

V² = U² + 2as

2as = V² – U²

as = V² – U²/2

as = V²/2 – U²/2**But as the value in W = mas**

W = m(V²/2 – U²/2)**Multiply m in bracket**

W = mV²/2 – mU²/2**Arrange the half in first**

W = 1/2mV² – 1/2mU²**Kinetic energy Formula = 1/2mV²**

so W = Kf – Ki

Hence, Proved

## Work Energy Theorem for Variable Force

W = mas**Both Side Integration**

∫dw = ∫mads \ a = dv/dt

w = mdv/dt.ds**Replace s with v**

w = mds/dt.dv \ ds/dt = v

w = ∫vdv

w = m [v²/2] **Integral v and u**

w = m [ v²/2 – u²/2]

w = 1/2mv² – 1/2mu²

w = Kf – Ki

Hence, Proved

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