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Classical Mechanics

# Potential Energy: Level 2-3 Challenges

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 of $$N$$ particles whose coordinates are $$\mathbf{r}_i = \{r_i^x,r_i^y,r_i^z\}$$, and whose velocities are $$\dot{\mathbf{r}}_i$$.

Each pair of particles $$\left(i, j\right)$$ interacts through a potential $$V\left(\mathbf{r}_i, \mathbf{r}_j\right)$$ which depends on the coordinates $$\left(\mathbf{r}_i, \mathbf{r}_j\right)$$ only through their difference, i.e. $V\left(\mathbf{r}_i, \mathbf{r}_j, t\right) = V\left(\mathbf{r}_i-\mathbf{r}_j, t\right)$ As the system evolves in time, which of the following bulk quantities must be conserved?

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