The main use of first order differential equations is in modeling how different aspects of our world change over time. This class will be spent as a problem solving session to try to get used to modeling. In particular, we will try to solve the following two problems:

A Leaky Container

For a cylindrical can of radius $r$ with a small hole punched in the bottom, water can escape through this hole at a rate $\pi r^2\sqrt{2gh(t)}$ where $h$ is the height of the water at time $t$. Suppose we take a standard pop can ($V=355ml$, $r=3.84 cm$) and add water at a constant rate $10ml/s$. Will the water overflow from the can?

Rocket Motion

In straight-line motion, we can use Newton's second law to write the acceleration of an object as the sum of the forces on the object. However, if the mass of the object is changing over time, then we can rephrase Newton's second law to be in terms of a change in an object's momentum. Thus $$ \frac{d}{dt} (mv) = \frac{dm}{dt}v+m\frac{dv}{dt} = F.$$ By conservation of momentum, we can rewrite this as $$ m \frac{dv}{dt} = F + u\frac{dm}{dt},$$ where $u$ is the veloctity of the lost (or added) mass relative to the object.

Suppose a rocket has mass $M$ when empty and is carrying $2M$ worth of fuel. If $\frac{dm}{dt} = -k$ while the rocket has fuel, $v(0) = 0$, spent fuel is ejected at a constant speed $u$, and the only force acting on the rocket is the force of gravity $(F=-mg)$. Find the maximum height of the rocket.