# Principle of Conservation of Energy

- The energy is a
**conserved quantity**. There won’t be any change in the energy of an**isolated system**. - The energy
**can neither be created**nor can it be destroyed. It can only be**changed from its one**form to another form. - For example: if we throw a stone upward with certain velocity, initially, it
**has maximum velocity**and hence**maximum kinetic energy.** - As it moves up, its
**velocity goes on**At a certain height, its**velocity becomes**zero, and hence**its kinetic energy becomes**zero. **At the same**time, the stone gains height as it moves up.- As the stone gains height, its
**potential energy**increases. Hence, it is clear that**when a stone is**thrown up, there is continuous decrease in kinetic energy and increase in the potential energy. - However,
**the total energy**of the stone**remains the same**. This is the**principle of conservation of energy.** **According to principle**of conservation of energy, the energy can neither be created nor be destroyed but can be changed from one form to another form.

Image Source: nuclearpower Image Source: Braincart

**Energy conservation in free fall**

Consider a body of mass **m **is initially at **a point A** which is at a height **h** from the ground as in the given figure. Let the body fall freely under gravity so that the acceleration of the body is **g**– acceleration due to gravity. After a certain time, the body reaches a **point B** which is at **height (h-x)** from the ground.

As the body falls down, its velocity increases. Finally, the body strikes the **ground C** with velocity v.

**For the point A**

When the body is at A, velocity **(u) = 0** and the height from the ground **(h) = h.**

Hence, the kinetic energy of the body **(K.E) _{a} = ½ mu^{2}** = 0

And the potential energy of the body is **(PE) _{a} = mgh**

The total mechanical energy of body at point A is given by

E_{a }= (K.E)_{a} + (P.E)_{a} = 0 + mgh = mgh

**∴ E _{a} = mgh** …………………………………………………………. (i)

**For the point B**

When the body is at point B on its way down, we have initial velocity **(u) = 0**

Final velocity **(v) = v _{b}**

Distance travelled = AB=AC – BC = h – (h-x) **= x**

We have, **v ^{2} = u^{2} + 2gh (formula)**

Or, v_{b}^{2} = 0 + 2gx = 2gx

**∴ v _{b}^{2} = 2gx**

Now, the kinetic energy of the body at point B is

(K.E)_{b }= ½ mv_{b}^{2 }= ½ m(2gx) = mgx

Also, the potential energy of the body at point B is

(P.E)_{b} = mgh = mg(h-x)

Therefore, the total mechanical energy of the body when it is at the **point B** is given by

**E _{b}** = (K.E)

_{b}+ (P.E)

_{b}= mgx + mg(h-x) = mgx + mgh – mgx = mgh

**∴ E _{b}= mgh**…………………………………………………………… (ii)

**For the point C**

When the body is at point C, we have, for the motion of the body

Initial velocity **(u) = 0**

Final velocity **(v) = v _{c}**

Distance travelled **= h**

We have, **v ^{2}= u^{2} + 2gh**

or, v_{c}^{2 }= 0 + 2gh =2gh

**∴v _{c}^{2 }= 2gh**

Hence, when the body is at point C, the kinetic energy is given by,

(K.E)_{c} = ½ mv_{c}^{2} = ½ m(2gh) = mgh

Similarly, the potential energy of the body at the point C is given by,

(P.E)_{c} = mg(height of point C) = mg × 0 = 0

Hence, the total mechanical energy of the body when it is at the point C is given by,

E_{c} = (K.E)_{c} + (P.E)_{c} = mgh + 0 = mgh

**∴ Ec = mgh**………………………………………………………… (iii)

**From equation (i), (ii) and (iii),** it is seen that the total mechanical energy of a freely falling body when it is at point A is equal to the energy when it is at B or C.

**This implies** that the total mechanical energy of a **freely falling body is conserved**.

**In other words**, the principle of conservation of energy **holds true** in this case of a freely falling body.

**If a graph is plotted** between the total mechanical energy of a freely falling body and its height from the ground, **a curve is obtained** as in figure below.

**References: **

i) https://examples.yourdictionary.com/law-of-conservation-of-energy-examples.html

ii) https://web.fscj.edu/Milczanowski/psc/lect/Ch3/slide5.htm

iii) https://www.grc.nasa.gov/www/k-12/airplane/thermo1f.html

## Principle of Conservation of Energy