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

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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