Conservation of Energy Example

During a flood a tree trunk of mass \(\text{100}\) \(\text{kg}\) falls down a waterfall. The waterfall is \(\text{5}\) \(\text{m}\) high.

If air resistance is ignored, calculate:

• the potential energy of the tree trunk at the top of the waterfall.

• the kinetic energy of the tree trunk at the bottom of the waterfall.

• the magnitude of the velocity of the tree trunk at the bottom of the waterfall.

Step 1: Analyse the question to determine what information is provided

• The mass of the tree trunk \(m = \text{100}\text{ kg}\)

• The height of the waterfall \(h = \text{5}\text{ m}\).

  These are all in SI units so we do not have to convert.

Step 2: Analyse the question to determine what is being asked

• Potential energy at the top

• Kinetic energy at the bottom

• Velocity at the bottom

Step 3: Calculate the potential energy at the top of the waterfall.

\begin{align*} {E}_{P} & = mgh \\ & = \left(\text{100}\text{ kg}\right)\left(\text{9.8}\text{ m·s$^{-2}$}\right)\left(\text{5}\text{ m}\right) \\ & = \text{4 900}\text{ J} \end{align*}

Step 4: Calculate the kinetic energy at the bottom of the waterfall.

The total mechanical energy must be conserved.

\[{E}_{K1} + {E}_{P1} = {E}_{K2} + {E}_{P2}\]

Since the trunk's velocity is zero at the top of the waterfall, \({E}_{K1}=0\).

At the bottom of the waterfall, \(h = \text{0}\text{ m}\), so \({E}_{P2}=0\).

Therefore \({E}_{P1} = {E}_{K2}\) or in words:

The kinetic energy of the tree trunk at the bottom of the waterfall is equal to the potential energy it had at the top of the waterfall. Therefore \({E}_{K} = \text{4 900}\text{ J}\)

Step 5: Calculate the velocity at the bottom of the waterfall.

To calculate the velocity of the tree trunk we need to use the equation for kinetic energy.

\begin{align*} {E}_{K} & = \frac{1}{2}m{v}^{2} \\ \text{4 900} & = \frac{1}{2}\left(\text{100}\text{ kg}\right)\left({v}^{2}\right) \\ 98 & = {v}^{2} \\ v & = \text{9.899...}\text{ m·s$^{-1}$} \\ v & = \text{9.90}\text{ m·s$^{-1}$} \end{align*}