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# Conservation of Energy

Energy cannot be created or destroyed in any transformation. This powerful accounting principle helps us analyze everything from particle collisions, to the motion of pendulums.

Suppose a STOL plane's landing speed is 45 m/s (\(\approx\) 100 mph). Neglecting any other effects like wind flaps, or drag, how small will the plane's velocity (in m/s) be at the top of the runway?

**Details**

- The runway itself is 450 m long, i.e. if you walked from the bottom to the top, you'd walk 450 m along the runway.
- An \(f\)% incline means that if you walk a distance \(d\) along an incline, your rise is given by \(fd/100\).

Climbing out of bed is sometimes hard in the morning. If you have a mass of \(60~\mbox{kg}\), what is the minimum amount of work **in joules** you have to do to get out of bed? Assume your center of mass when lying in your bed is \(0.75~\mbox{m}\) above the floor, and your center of mass when standing is \(1.25~\mbox{m}\) above the floor.

**Details and assumptions**

- The acceleration due to gravity is \(-9.8~\mbox{m/s}^2\).

A particle is constricted to move along the positive \(x\)-axis under the influence of a potential energy:

\[U(x) = \frac{3}{x} + 7x\]

Find the point of equilibrium for the particle. (Round your answer to three decimal places.)

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