FAQ - How to solve these problems
There are (and will be) different problems from various parts of Math and Physics. Usually the problem gives you some starting information and asks to deduce some additional information that is not explicitly given in the problem itself, but is somehow hidden in the initial data and through some logical steps can be revealed. Sometimes the problem has only one solution, sometimes it can have many, sometimes none.

Often problems are very particular: you are given some numbers, you add or subtract them and get the answer. Usually, that type of problems are the easiest to solve (relatively). Sometimes, you can even guess the answer, without even going through all the steps of the solution. That's cool, often this method can be the fastest way to solve the problem and therefore it is a very practical method. I would call it an applied method or an engineering method, in a sense that the solution to the problem is very particular and can not be reused in the future.

On the other side, there is a more abstract or theoretical method of a solution, when you solve the problem in a general form and only then apply that general solution to a given problem. I believe that science is first of all about the second method, as science is not so much concerned with solutions to some particular problems but rather is trying to uncover the relations between seemingly unrelated things and describe these relations in the language of mathematics. That is usually called a fundamental science. In practice every solution of every problem we encounter has some elements of both the applied and the theoretical methods. Many particular problems that are on this website could be solved by different methods, however, while a general solution might be more difficult to figure out, I consider it to be a more scientific solution and therefore value it higher than a fast but less general solution. I will give two examples of that (from Math and Physics).

1. Let's say you are asked to solve x+2=5 equation.
You can almost instantly see that the correct answer is x=3. That is, however, a very particular solution and it will not anyhow help you to solve any other equation.

To solve it in a more general form you can rewrite it in a form x+a=b and then say that x=b-a. This solution is more general than the first one, since it can be applied to different "a" and "b".
You can go even further and make it more general if you like.

2. Imagine now that the problem asks to find the time it takes for a ball to fall to the ground from the height of 4.9 meters above ground.
One solution would be to use the formula H=(g*t^2)/2 => t = square_root(2H/g) = square_root(2*4.9/9.8) = 1 second.

Again, while the solution is correct, you can easily make it more general, by assuming that ball is falling from some arbitrary height H and the original formula will no longer be true (since g will not be constant any more).
And again, you can go further: assume the ball and the planet where its falling to (if it's a planet) have electric charges, assume there is air drag, assume the ball has some horizontal speed, etc.

Since a general solution is valued more than a particular, even if the problem has been already solved, it is often possible to solve it in a more general way or by using a more efficient method or a different approach.

Therefore, even if the problem seems to be overly simple or you already know the solution, etc., do not just write it off immediately. Think about it for a moment, and who knows, maybe one apple that falls from the tree and hits you in the head will lead to wonderful conclusions.

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