Problems

01

Pursuit with a crosswind

An airplane starts at $(a,0)$ and flies to the origin $(0,0)$, but there is a crosswind. The wind blows in the $+y$ direction at a constant speed $v_w$, and the airplane flies with a fixed speed $v_a$ relative to the air around it. The plane always flies with its fuselage pointing towards $(0,0)$. Assume $v_a>v_w$. Find the path taken by the plane over the ground. (Hint: recall that $\frac{dy}{dx}=\frac{y^{\prime }(t)}{x^{\prime }(t)}$.)

02

Escape velocity

A body is launched towards space from the surface of the earth. Assume the force of gravity on the body is given by $-\frac{mg{R}^2}{(R+z{)}^2}$, where $R$ is the radius of the earth, $z$ is the altitude of the body above the surface of the earth, and $m$ and $g$ are constants representing the mass of the body and the gravitational constant. Ignore air resistance. What is the minimum launch velocity $v_0$ needed for the body to escape earth’s gravity?

• Using Newton’s Law, write down a differential equation satisfied by the body during flight.
• Write this equation in terms of the velocity $v$ (dependent) and altitude $z$ (independent). (You will need to eliminate $dt$ using the chain rule.)
• Solve this equation by separating variables, writing an explicit function $v(z)$. Use the initial condition $v(0)=v_0$.
• Find the maximum altitude in terms of $v_0$. (Find $z$ when $v=0$.) Now find $v_0$ which produces an infinite maximum altitude.

03

Coffee cooling

Newton’s Law of Cooling says that an object at temperature $T$, when placed in an ambient environment at temperature $T_A$, will cool according to the following differential equation:

$$\frac{dT}{dt}=-k(T-T_A),$$

where $k$ is some constant that depends on the properties of the object and environment.

Suppose you have a cup of hot black coffee and some creamer at room temperature. You are going to drink the coffee in 15 minutes, and you want it to be as hot as possible. Do you put the creamer in now, or wait until just before drinking?

(You should assume that the temperature taken on by a mixture of coffee and creamer is the average of their original temperatures, weighted by their original quantities.)