Classical Mechanics
# Mechanics Warmups

The distance from Boston to San Francisco is 4344 km. However, a flight from Boston to San Francisco is about 6.33 hours of flying time (the flying time is the total time minus the time required to takeoff and land), whereas a flight from San Francisco to Boston is only 5.33 hours of flying time.

The difference is due to the jet stream, which blows across the United States roughly from west to east. If the speed of the airliner relative to the air is a constant, and the jet stream is considered as blowing at a constant speed directly against the airplane as it flies from Boston to San Francisco and directly with the airplane on the return flight, then what is the speed of the jet stream **in km/hour**?

The average human heart pumps $5~\mbox{L/min}$ of blood through the human body at a pressure of $10^4~\mbox{Pa}$. For a person who lives 80 years, what is the total energy output of his heart **in Joules**?

**Details and assumptions**

- Assume that the cardiac output is constant for any age.
- There are 365.25 days in one year, and 24 hours in a day.

Josh makes a cup of coffee at a station that is a few minutes' walk from his lab. The coffee is piping hot when it fills the cup and he pours soy creamer into the coffee at some point before he drinks it. The creamer is kept in a refrigerator until it is used and is much colder than the coffee.

If Josh wants the coffee to be as hot as possible when he gets back to the lab, at what point should he add the creamer to the coffee?

One of the most lethal NERF guns is the Strongarm, a 6 shooter revolver known for its rapid fire. The firing mechanism is straightforward, a spring is used to drive air through a cylinder which then forces the dart out of the barrel.

Its popularity is due in part to the ease with which it can be modified to shoot faster and farther. One easy modification is to insert a secondary spring inside the main spring, as shown below:

If the strength (spring constant) of the first spring is $k$, and the strength of the second spring is $\frac12 k$, what is the strength of the two springs together?

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 $V.$