When teaching implicit differentiation in freshman calculus I lack good examples which might help students relate the theory to applications in other sciences.

So I'm looking for (relatively simple) examples from physics/engineering/life science/economics etc. of two quantities $x,y$ that depend on each other in such a way, that neither $x$ is a function of $y$, nor $y$ is a function of $x$. In other words: $x,y$ should fulfil some equation, such that there exists an $x$ value for which the equation has at least two solutions in $y$ and vice versa.

Equations are great, but if you have some examples where the relation between $x$ and $y$ is not easily expressed by an equation, but instead by a graph or some other data that is also fine.

My students are mainly first year engineers.

When teaching implicit differentiation in freshman calculus I lack good examples which might help students relate the theory to applications in other sciences.

So I'm looking for (relatively simple) examples from physics/engineering/life science/economics etc. of two quantities $x,y$ that depend on each other in such a way, that neither $x$ is a function of $y$, nor $y$ is a function of $x$. In other words: $x,y$ should fulfil some equation, such that there exists an $x$ value for which the equation has at least two solutions in $y$ and vice versa.

Equations are great, but if you have some examples where the relation between $x$ and $y$ is not easily expressed by an equation, but instead by a graph or some other data that is also fine.

My students are mainly first year engineers.

Edit: There is a related question here, but it's aimed for advanced analysis courses and doesn't ask for examples from applied areas. It has at least one applicable answer with isolines, but to explain it with equations in the firs semester single variable calculus, I would probably have to introduce functions of several variables.

When teaching implicit differentiation in freshman calculus I lack good examples which might help students relate the theory to applications in other sciences.

So I'm looking for (relatively simple) examples from physics/engineering/life science/economics etc. of tow quantities $x,y$ that depend on each other in such a way, that neither $x$ is a function of $y$, nor $y$ is a function of $x$. In other words: $x,y$ should fulfil some equation such that there exists an $x$ value for which the equation has at least two solutions in $y$ and vice versa.

Equations are great, but if you have some examples where the relation between $x$ and $y$ is not easily expressed by an equation, but instead by a graph or some other data that is also fine.

My students are mainly first year engineers.

When teaching implicit differentiation in freshman calculus I lack good examples which might help students relate the theory to applications in other sciences.

So I'm looking for (relatively simple) examples from physics/engineering/life science/economics etc. of two quantities $x,y$ that depend on each other in such a way, that neither $x$ is a function of $y$, nor $y$ is a function of $x$. In other words: $x,y$ should fulfil some equation, such that there exists an $x$ value for which the equation has at least two solutions in $y$ and vice versa.

Equations are great, but if you have some examples where the relation between $x$ and $y$ is not easily expressed by an equation, but instead by a graph or some other data that is also fine.

My students are mainly first year engineers.