Students will learn how to analyze and solve more complicated problems involving energy conservation.

### Key Equations

\begin{align*} \sum E_{\text{initial}} = \sum E_{\text{final} }\end{align*} ; The total energy does not change in closed systems

\begin{align*}KE = \frac{1}{2} \ mv^2\end{align*} ; Kinetic energy

\begin{align*}PE_g = mgh\end{align*} ; Potential energy of gravity

\begin{align*}PE_{sp} = \frac{1}{2} \ kx^2 \end{align*}; Potential energy of a spring

\begin{align*}W = F_{x} \Delta x = Fd \ \cos \theta\end{align*} ; Work is equal to the distance multiplied by the component of the force in the direction it is moving.

Guidance

The main thing to always keep prescient in your mind is that the total energy before must equal the total energy after. If some energy has transferred out of or into the system via work, you calculate that work done and include it in the energy sum equation. Generally work done by friction is listed on the 'after' side and work put into the system, via a jet pack for example, goes on the 'before' side. Another important point is that on turns or going over hills or in rollercoaster loops, one must include the centripetal motion equations -for example to insure that you have enough speed to make the loop.

#### Example 1

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### Explore More

- A rock with mass \begin{align*}m\end{align*} is dropped from a cliff of height \begin{align*}h\end{align*}. What is its speed when it gets to the bottom of the cliff?
- \begin{align*}\sqrt{mg}\end{align*}
- \begin{align*}\sqrt{ \frac{2g}{h} }\end{align*}
- \begin{align*}\sqrt{2gh}\end{align*}
- \begin{align*} gh\end{align*}
- None of the above

- In the picture above, a \begin{align*}9.0 \;\mathrm{kg}\end{align*} baby on a skateboard is about to be launched horizontally. The spring constant is \begin{align*}300 \;\mathrm{N/m}\end{align*} and the spring is compressed \begin{align*}0.4 \;\mathrm{m}\end{align*}. For the following questions, ignore the small energy loss due to the friction in the wheels of the skateboard and the rotational energy used up to make the wheels spin.
- What is the speed of the baby after the spring has reached its uncompressed length?
- After being launched, the baby encounters a hill \begin{align*}7 \;\mathrm{m}\end{align*} high. Will the baby make it to the top? If so, what is his speed at the top? If not, how high does he make it?
- Are you finally convinced that your authors have lost their minds? Look at that picture!

- When the biker is at the top of the ramp shown above, he has a speed of \begin{align*}10 \;\mathrm{m/s}\end{align*} and is at a height of \begin{align*}25\;\mathrm{m}\end{align*}. The bike and person have a total mass of \begin{align*}100 \;\mathrm{kg}\end{align*}. He speeds into the contraption at the end of the ramp, which slows him to a stop.
- What is his initial total energy? (Hint: Set \begin{align*}U_g = 0\end{align*} at the very bottom of the ramp.)
- What is the length of the spring when it is maximally compressed by the biker? (Hint: The spring does
*not*compress all the way to the ground so there is still some gravitational potential energy. It will help to draw some triangles.)

- An elevator in an old apartment building in Switzerland has four huge springs at the bottom of the shaft to cushion its fall in case the cable breaks. The springs have an uncompressed height of about \begin{align*}1\end{align*}meter. Estimate the spring constant necessary to stop this elevator, following these steps:
- First, guesstimate the mass of the elevator with a few passengers inside.
- Now, estimate the height of a five-story building.
- Lastly, use conservation of energy to estimate the spring constant.

- You are skiing down a hill. You start at rest at a height \begin{align*}120 \;\mathrm{m}\end{align*} above the bottom. The slope has a \begin{align*}10.0^\circ\end{align*} grade. Assume the total mass of skier and equipment is \begin{align*}75.0 \;\mathrm{kg}\end{align*}.
- Ignore all energy losses due to friction. What is your speed at the bottom?
- If, however, you just make it to the bottom with zero speed what would be the average force of friction, including air resistance?

- Two horrific contraptions on frictionless wheels are compressing a spring \begin{align*}(k = 400 \;\mathrm{N/m})\end{align*} by \begin{align*}0.5 \;\mathrm{m}\end{align*} compared to its uncompressed (equilibrium) length. Each of the \begin{align*}500 \;\mathrm{kg}\end{align*} vehicles is stationary and they are connected by a string. The string is cut! Find the speeds of the vehicles once they lose contact with the spring.
- A roller coaster begins at rest \begin{align*}120 \;\mathrm{m}\end{align*} above the ground, as shown. Assume no friction from the wheels and air, and that no energy is lost to heat, sound, and so on. The radius of the loop is \begin{align*}40 \;\mathrm{m}\end{align*}.
- If the height at point G is 76 m, then how fast is the coaster going at point G?
- Does the coaster actually make it through the loop without falling? (Hint: You might review the material from centripetal motion lessons to answer this part.)

#### Answers to Selected Problems

- .
- a. \begin{align*}2.3 \;\mathrm{m/s}\end{align*} c. No, the baby will not clear the hill.
- a. \begin{align*}29,500 \;\mathrm{J}\end{align*} b. Spring has maximum compressed length of \begin{align*}13 \;\mathrm{m}\end{align*}
- .
- a. \begin{align*}48.5 \;\mathrm{m/s}\end{align*} b. \begin{align*}128 \;\mathrm{N}\end{align*}
- \begin{align*}0.32 \;\mathrm{m/s}\end{align*} each
- a.29 m/s b. just barely, \begin{align*} a_C = 9.8 \;\mathrm{m/s}^2 \end{align*}