Classical Mechanics
# Conservation of Energy

Lukla airport in Nepal is one of the strangest in the world. Built to support tourism to the Himalayas, the airport has a single landing runway. What is more, the runway is only 20 m wide, 450 m long, and a 2,800 m cliff at the runway's end, leaving little room for error. In fact, the airport can only be used by so-called Short Takeoff and Landing planes (STOL). Helping somewhat is a 12% incline in the runway from start to finish, so that planes rise through over the course of their deceleration.

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 \(-9.8~\mbox{m/s}^2\).

Consider a system of \(N\) particles whose coordinates are \(\mathbf{r} = \{r_i^x, r_i^y, r_i^z\}\), and whose velocities are \(\{\dot{\mathbf{r}}_i\}\).

Each pair of particles \(\left(i, j\right)\) interacts through a potential \(V\left(\mathbf{r}_i, \mathbf{r}_j\right)\) which has no direct dependence on time. As the system evolves in time, which of the following bulk quantities must be conserved?

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.)

Consider a system with \(N\) particles whose coordinates are \(\mathbf{r}_i = \{r_i^x,r_i^y,r_i^z\}\), and whose velocities are \(\dot{\mathbf{r}}_i\). The energy of the system is described by the kinetic energy \(\frac{1}{2}m\sum \dot{\mathbf{r}}^2\) and an **effective** potential energy term \(V\left(\mathbf{r}_1, \ldots, \mathbf{r}_N\right).\)

All we know about \(V\) is that it depends on the positions only through their differences \(\mathbf{r}_1 - \mathbf{r}_2\), i.e. \[V(\{\mathbf{r}^i\}, t) = V\left(\mathbf{r}_1-\mathbf{r}_2, \mathbf{r}_1-\mathbf{r}_3, \ldots, t\right).\] As this system evolves in time, which of the following bulk quantities must be conserved?

**Assumptions**

- All interactions of the system with the outside world are described by \(V.\)

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