How should one manage knowns vs unknowns when solving a problem?

Question

I define a known as a variable with a number you can plug in. I define an unknown as a variable without a number you can plug in because none is given.

Recently I have taken several classes where they ask me to use formulas to compute some number. But it isn't just plug in and mindlessly compute. I have to be aware of what information I do not have and what formulas are usable.

For example,

Suppose I am considering a projectile motion problem in physics. The projectile is launched at a $45°$ angle and lands at a point $P$. Now I want to repeat the experiment but this time choose an angle $\theta$ such that the projectile bounces once with a perfectly elastic collision and lands at the same point. Visually this looks like harmonics with the first throw being $f$ and the second being $2f$, $f$ standing for frequency.

Note how I am not given information about $t$ or $v_{\text{ball}}$. Now, I may be able to use formulas to determine a bit more. For instance, I found

$$t_{full}= \frac{2v_b \sin(45°)}{g}$$

But this formula is still in terms of a variable I have no value for. And for this particular problem, I basically went in circles because I couldn't organize what I knew and didn't know to find an arrangement of my knowns and unknowns that would a numerical result.

In such problems, many formulas look like they are applicable. But in reality, you cannot use a formula if you don't have stuff to plug into it.

My Question

Is there an efficient way to keep track of what values you have and do not have and use this to determine what formulas are worth considering?

Remark:

I read a book on algorithms once that suggested breaking a large problem down into sub-problems and solving them to solve the larger problem. I think something like that is what I am looking for, but more like a notation specifically for solving problems that makes that subdivision easier to do.

Answer 1

Since it is difficult to talk only in the abstract, let's actually write down a solution to this problem.

I assume that the problem intends the initial velocity $v_0$ of the ball to be held fixed.

First of all, what are you trying to achieve by changing $\theta$ from $45^{\circ}$ to something else? What you want is for the horizontal displacement $d$ of the ball when it touches down to be half as large as it was before. So it is natural to write down the relationship between $d$ and $\theta$. Of necessity, $v_0$ will intervene in this calculation. (The faster the projectile is launched, the further it will go, if $\theta$ is held constant.) There are no other variables. The angle $45^{\circ}$ is just one possible value of $\theta$, so in doing this calculation you should not assume that $\theta = 45^{\circ}$ at first, although that will be one possibility you will look at.

The result is that $$d = \frac{v_0^2}{g} \sin 2\theta$$

So the question is how to change $\theta_1 = 45^{\circ}$ to an angle $\theta_2$ so that $d$ is halved. We have $$d_1 = \frac{v_0^2}{g} \sin 2\theta_1 = \frac{v_0^2}{g}$$ and $$d_2 = \frac{v_0^2}{g} \sin 2\theta_2.$$

At this point, you have done the necessary calculations of $d_1$ and $d_2$ in terms of the values of $\theta$, and you must ask yourself what condition you wish to satisfy. That condition is that $d_2/d_1$ should be $1/2$. Writing down the corresponding equation, you find $$\frac{(v_0^2/g)\sin 2\theta_2}{v_0^2/g} = \frac{1}{2},$$ which simplifies to $$\sin 2 \theta_2 = 1/2.$$

This equation shows that there are two solutions, namely $\theta = 15^{\circ}$ and $\theta = 75^{\circ}$.

Initially, it might have seemed as though the answer about how you should change $\theta$ would depend in some way on $v_0$. From a strictly logical standpoint, there was nothing that allowed you to assume that this would not be the case. It was only a mathematical calculation that allowed you to determine this (although it might be possible to give some kind of physical argument).

Let's summarize the status of the different variables in this problem. $\theta$ was the variable you were required to change, so that a prescribed change would correspondingly be obtained in the variable $d$. You computed $d$ in terms of $\theta$ (since $d$ is the result of a certain physical process that can be described when $\theta$ is known.) We could say that $\theta$ was an independent variable and $d$ the dependent variable. The important thing is that the physical nature of the problem makes it such that $d$ can be effectively calculated provided that you temporarily pretend that $\theta$ is known, even though it isn't really.

In order to perform this calculation of $d$, however, in truth you also required knowledge of $v_0$. We say that $v_0$ is a parameter. It is a sort of background variable which you must regard as known even though it may not really be. (The number $g$ could also be regarded as a parameter, especially if you were going to perform the experiment on different planets, for example. In the problem as stated, that is beside the point.)

Since you were required to find $\theta$, the key step was to write down the equation entirely in terms of $\theta$ (and any parameters), rather than by using the variable $d$.

Initially it seemed that the answer to the question might depend on the parameter $v_0$, but eventually the calculations showed that this was not the case.

The main point to consider in determining your variables was which ones were needed in order to compute $d$. This is really a physics question.

Answer 2

As others have commented, this situation is a little more general than just in mathematics. In my view, it borders on issues on mathematical modelling and general engineering rather than just mathematics, because one has to decide how to "adjust the situation" and "decide what constitutes a solution", rather than just executing an algorithm to compute a numerical solution to a well-understood problem with well-understood methods.

However, one can tackle it mathematically anyway, and in a way that can be useful for those that come after. The primary idea: DOCUMENT YOUR ASSUMPTIONS.

You may have some rational for picking a particular set of tools to tackle a problem. It is a good idea to write down your rationale for your choice of tools, and don't be afraid to say that you don't know any better methods (yet), but even if you are compelled to use a limited tool box, you will run into cases where you can't use them automatically because of the unknowns that you encounter, e.g. your tool needs four inputs and you only have three.

I recommend organizing your data by tools and by inputs that you know or easily understand and simulate. Inputs that you don't know you treat as variables, and if the methods you use are well understood, your initial result is a nicely-expressed function or relation in the unknown variables which you can analyze. Depending on the goal, the problem might be finishable as you might be looking for an optimal answer: you can then conclude that the most optimal solution implies certain values for the unknowns, or that a certain relation must hold between the unknowns.

The point is that by documenting your process and assumptions, others can replicate it or at least approximate it and check your work. Even if your work does not represent the last word on the matter, it might be a good initial point on which others will base their analyses. However, they can't do it if you don't tell them. I.E., Show Your Work!

Gerhard "Your First-Grade Teacher Was Right" Paseman, 2015.06.22