# Conservation of Mechanical Energy

What is conservation of mechanical energy ?

- In an isolated system where only conservative forces cause changes, the kinetic energy and potential energy can change , but their sum , the mechanical energy
**E**of the system , cannot change .

Theory :If the kinetic energy isKand the potential energy isUof an object then,

$K_{i}$ + $U_{i}$ = $K_{f}$ + $U_{f}$

where ; the subscriptsiandfrefer to the initial and final states of the system.

-Proof :When a conservative force does workWon an object within the system , that force transfers energy between kinetic energyKof the object and potential energyUof the system . FromWork-Kinetic energy theorem, we can write ,

$\Delta$ K = $K_{f}$ - $K_{i}$ = W

And fromWork-Potential energy theorem, we can write,

$\Delta$ U = - W

Now , combining this two equation , we find that ,

$\Delta$ K = - $\Delta$ U

We can rewrite this equation as ,

$K_{f}$ - $K_{i}$ = - ( $U_{f}$ - $U_{i}$ )

so, $K_{f}$ - $K_{i}$ = $U_{i}$ - $U_{f}$

By rearranging this equation we find that ,

$K_{i}$ + $U_{i}$ = $K_{f}$ + $U_{f}$

**Example :**A boy of mass 50 kg is released from rest at the top of a water slide , at height h=10 m above the bottom of the slide . Assuming that the slide is frictionless because of the water on it . Find the boy's speed at the bottom of the slide.**Answer :**We have ,

**$K_{i}$ + $U_{i}$ = $K_{f}$ + $U_{f}$**

or, $\frac{1}{2}$m$v_{f}^{2}$ + mg$h_{f}$= $\frac{1}{2}$m$v_{i}^{2}$ + mg$h_{i}$

Dividing by m and rearranging yield ,

$v_{f}^{2}$ = $v_{i}^{2}$ + 2g( $h_{i}$ - $h_{f}$ )

Putting $v_{i}$ = 0 and ( $h_{i}$ - $h_{f}$ ) = h leads to ,

$v_{f}$ = $\sqrt{2gh}$

= $\sqrt{2 \times 9.8 ms^{-2} \times 10 m}$

= 14 m/s**Try it yourself :**A block of mass m = 2.0 kg is dropped from height h = 40 cm onto a spring of spring constant k= 1960 N/m . Find the maximum distance , the spring is compressed ? Note that , for a spring , U = $\frac{1}{2}$k$x^{2}$ .**Experiments :**

**Cite as:**Conservation of Mechanical Energy.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/conservation-of-mechanical-energy/