## Conservation of energy

The law of conservation of energy states that the total energy in an isolated system is constant. Energy is not created or destroyed; it changes form. For our purposes, total energy is the sum of kinetic and potential energies.

When a ball is thrown upwards, it is gaining potential energy and losing kinetic energy joule for joule. When it starts falling back down, the trade occurs in reverse. Mathematically,

Delta E_"k" + Delta E_"g" = 0.

When we substitute those energy changes, we can rearrange to get

v_2^2 - v_1^2 = -2gDelta h.

This looks very similar to the eighth and final equation that we derived in the section on the equations of motion of the first unit.

If we were instead considering a situation where kinetic energy was converted to elastic energy (a ball hits a spring and compresses it), our equation would be Delta E_"k" + Delta E_"e" = 0.

### Example

A 2.5 kg block is dropped from 7.5 m above the floor. What is its speed as it hits the floor?

We can used the equation that I just mentioned:

v_2^2 - v_1^2 = -2gDelta h.

Solving for v_2, we have

v_2 = sqrt(v_1^2-2gDelta h) = sqrt(0-2(9.80\ "N/kg")(- 7.5\ "m")) ~~ 12.124,

therefore the final speed is 12 m/s.