Students will learn about kinetic energy, how and when to apply it and how to use kinetic energy

### Key Equations

**Kinetic energy**

\begin{align*} K = \frac{1}{2}mv^2 \begin{cases} m & \text{mass~(in~kilograms,~kg)}\\ v & \text{speed~(in~meters~per~second,~}\text{m}/\text{s}\text{)} \end{cases}\end{align*}

#### Example 1

You are using a sling to throw a small stone. If the sling is .5 m long and you are spinning it at 15 rad/s, how high would the rock go if you throw it straight up?

##### Solution

We'll start by setting the kinetic energy of rock to it's gravitational potential energy at it's maximum height and then solving for the rock's height.

\begin{align*} KE_i&=PE_f\\ \frac{1}{2}mv^2&=mgh\\ h&=\frac{v^2}{2g}\\ \end{align*}

We still don't know the rock's linear velocity, but we do know the sling's angular velocity and radius so we can put those into the equation instead.

\begin{align*} h&=\frac{(\omega r)^2}{2g}\\ h&=\frac{(15\;\text{rad/s} * .5\;\text{m})^2}{2*9.8\;\text{m/s}^2}\\ h&=2.9\;\text{m}\\ \end{align*}

### Time for Practice

- A bomb with 8 x 104 J of potential energy explodes. Assume 20% of its potential energy is converted to kinetic energy of the metal pieces flying outward (shrapnel).
- What is the total kinetic energy of the shrapnel?
- Assume the average mass of the shrapnel is 0.4 kg and that there are 200 pieces. What is the average speed of one piece?

- A 1500 kg car starts at rest and speeds up to 3.0 m/s.
- What is the gain in kinetic energy?
- We define efficiency as the ratio of output energy (in this case kinetic energy) to input energy. If this car’s efficiency is 0.30, how much input energy was provided by the gasoline?
- If 0.15 gallons were used up in the process, what is the energy content of the gasoline in Joules per gallon?

- An airplane with mass \begin{align*}200,000 \;\mathrm{kg}\end{align*} is traveling with a speed of \begin{align*}268 \;\mathrm{m/s}\end{align*}.
- What is the kinetic energy of the plane at this speed?

A wind picks up, which causes the plane to lose \begin{align*}1.20 \times 10^8 \;\mathrm{J}\end{align*} per second.

- How fast is the plane going after \begin{align*}25.0\end{align*} seconds?

#### Answers to Selected Problems

- The answers are:
- \begin{align*} 1.2 \times 10^6 J \end{align*}
- \begin{align*} 20 m/s \end{align*}

- The answers are:
- \begin{align*}6750 J\end{align*}
- \begin{align*}2.25 \times 10^5 \;\mathrm{J}\end{align*}
- \begin{align*}1.5 \times 10^5 \;\mathrm{J/gallon \ of \ gas}\end{align*}

- a. \begin{align*}7.18 \times 10^9 \;\mathrm{J}\end{align*}