Are there any "related rates" calculus problems that don't feel contrived?
The skills that students are practicing in related rates problems are:
Differentiating a known equation implicitly with respect to time.
Interpreting the time derivative of a quantity as a rate of change.
The main reason that related rates problems feel so contrived is that calculus books do not want to assume that the students are familiar with any of the equations of science or economics. Every related rates problem inherently involves differentiating a known equation, and the only equations that the calculus book assumes are the equations of geometry.
Thus, you can find related rates problems involving various area and volume formulas, related rates problems involving the Pythagorean Theorem or similar triangles, related rates problems involving triangle trigonometry, and so forth. A few of these problems are compelling -- for example, computing the speed of an airplane based on ground observations of its altitude and apparent angular velocity -- but most of them do feel a bit contrived.
The reality, of course, is that students are familiar with many of the basic equations and concepts of science and economics, and there's no rule against using these in problems. For example, you can make up all sorts of compelling related rates problems by starting with any physics or chemistry equation and imagining a situation where you might want to take its derivative:
The kinetic energy of an object is $K = \frac{1}{2}mv^2$. If the object is accelerating at a rate of $9.8 \text{m}/\text{s}^2$, how fast is the kinetic energy increasing when the speed is $30 \;\text{m}/\text{s}$?
An ideal gas satisfies $PV = nRT$, where $n$ is the number of moles and $R \approx 8.314\;\; \text{J}\; \text{mol}^{-1} \text{K}^{-1}$. Give the rate at which the temperature and volume of the gas are increasing, and then ask about the rate of change in pressure when the volume and temperature reach certain amounts.
The total energy stored in a capacitor is $\frac{1}{2} Q^2 / C$, where $Q$ is the amount of charge stored in the capacitor and $C$ is the capacitance. Give the value of $C$ and the rate at which $Q$ is decreasing, and ask about the rate at which the capacitor is losing energy when the energy is a certain amount.
In astronomy, the absolute magnitude $M$ of a star is related to its luminosity $L$ by the formula $$ M \;=\; M_{\text{sun}} -\; 2.5\; \log_{10}(L/L_{\text{sun}}). $$ where $M_{\text{sun}} = 4.75$ and $L_{\text{sun}} = 3.839 \times 10^{26} \text{watts}$. (Note that, by convention, brighter stars have lower magnitude.) If the absolute magnitude of a variable star is decreasing at a rate of $0.09 / \text{week}$, how quickly is the luminosity of the star increasing when the magnitude is $3.8$?
It's easy to make these up: just think of any equation in science or economics whose derivative might be interesting. Wikipedia and/or science textbooks can be helpful for finding equations from a wide variety of fields.
Edit: I have compiled a list of these problems in the form that I use them in my classes, and posted them on my professional web page.
Here's one that may be quite relevant to their lives.
Doppler radar measures the rate of change of the distance from an object to the observer. A police officer $a$ metres from a straight road points a radar gun at a car travelling along the road, $c$ metres away, and measures a speed of $v$. What is the car's actual speed?
Here are two examples I think are interesting.
- A ladder that is leaning against a wall starts slipping down. If the point where the ladder touches the ground (draw your own picture) is moving away from the wall at a constant rate, is the point where the ladder touches the wall falling at a constant rate? People may reasonably think that because the ladder is "rigid" the answer should be yes.
Differentiating $x^2 + y^2 = L$ (where $L$ is the length of the ladder, a constant) shows $dx/dt$ being constant certainly doesn't make $dy/dt$ constant, and in fact since $dy/dt = -x(dx/dt)/\sqrt{L-x^2}$ we see that as $x$ increases (while being less than $L$) and $dx/dt$ is fixed the point where the ladder touches the wall is dropping faster and faster. Some students may even say that from their experience or physical intuition this actually makes sense, which raises the question of whether this is truly a physical phenomenon or a purely mathematical one that has been revealed from calculus.
- There is a gas is in a chamber with a flexible wall (so the chamber can expand or contract, e.g., a piston is at one end). According to the chemists, if we maintain the gas at a constant temperature while increasing or decreasing the size of the chamber then the pressure $P$ and volume $V$ satisfy the relation $PV$ = constant. (I am thinking of the ideal gas law $PV = nRT$, where $n$ and $T$ don't change.) One aspect of this equation which is obvious is that as the volume goes up/down, the pressure goes down/up. Question: If we decrease the volume at a constant rate, does the pressure increase at a constant rate, and more precisely at the rate which is the reciprocal of the rate at which the volume is going down? I think it's quite natural for people to make a snap judgement that if $dV/dt = -4$ then $dP/dt = 1/4$ because $PV$ is constant, but of course the product rule shows this is wrong, and moreover if $dV/dt$ is constant then $dP/dt$ is definitely not constant.
It's perhaps worth first discussing a situation where such intuition is right, namely where the sum of two variables is fixed, rather than the product. Find your own example where two variables $x$ and $y$ satisfy $x+y$ = constant. Then $dx/dt = -dy/dt$, which makes a lot of sense: the rate at which one goes up is exactly "opposite" to the rate at which the other goes down. But when the product is constant this is completely incorrect.