OK, we crossed in posting. Some of what I write below is already understood.
What you are asking for is very standard algebra. But you are clearly starting from a very early base in your understanding of mathematics. You should really do some reading as it isn’t something that can be instantly understood in the space of a few postings. But a few things that might help.
You have equations that describe the motion of things. You also have equations that describe desired conditions. There are rules that allow you to manipulate these equations, including combining them and breaking them up, and you can make use of these rules to create new equations that are your answer.
So, you have equations that describe the movement of the ball. In 2 dimensions you can have equations for its position written in a number of ways. A critical point is that all these equations are based upon time - as you change time the position changes.
You have equations that describe desired conditions. In this case you want to intercept the ball. This means you have an equation that says “at a given time, position of ball is the same as the position of the interceptor”.
So in very high level form:
Location_ball = location_function_ball(time)
Location_person = location_function_person(time)
So to intercept the ball with the person:
location_function_ball(time) = location_function_person(time)
That is the X = Y at time T Napier mentioned.
The basic form of any problem of this type is “solving the equation(s)”. Here we create set them all to the form: equation = 0. This is easy, since we just rearrange them to:
location_function_ball(time) - location_function_person(time) = 0
Where it gets harder is to create the equations that capture your particular problem. A common, and easiest version of the above is to ask “does the ball’s path meet the person.” This happens if you can solve the equation with a value of time that makes sense. If the location functions for the ball and the person are both fixed, you have a simple yes/no answer.
Your problem is more involved. You want to be able to vary the location function for the person in such a way that you guarantee that the interception occurs. For this you need to identify what parameters of the person’s location function you can vary. These become new unknowns in the equations. So, you have equations that have a number of variables, whereas in the simple case only time varied.
What you probably want to do is manipulate the equations so that you eliminate time. Ideally you then end up with something that is an equation involving your other free variable, and some constant expression. This is the answer. Set this free variable in the person location function to that value, and your person will intercept the ball.
This isn’t always possible. But for the simple system you describe it will be, unless you apply constraints such as maximum acceleration or maximum speed to the person intercepting the ball.
You may also discover that you have more than one solution. For instance if the ball takes a curved path you can intersect it at more than one place.
That was for one free variable in the person location function. However you have more than one. At least direction and speed, so two So you will find that you have a result where you have possibly an infinite number of solutions, you can intercept the ball at any place along it’s path by running in the appropriate direction and at the right speed. Clearly you probably want to run the shortest distance, so this becomes another constraint, and another equation.
You can imagine all of this graphically. For a 1D problem this is easy. One axis is time, the other the one dimension. Draw the path of the ball and the path of the person. Do they intersect? For a 2D game you need 3D graph paper, which is a bit hard to find. But the idea is there.
This is where you need to do some reading and spend some effort. I’ve slid over the mechanics of what is done to solve the system. Nothing is especially difficult, mostly involving the application of a set of rules as if they were a set of tools in a toolbox. If you can solve a Sokoku puzzle, you can do linear algebra. Where you could get more advanced is to explicitly look at the area of linear algebra. What you get here is a set of tools that simplify how you represent and manipulate the equations. Rather than longhand equations, you can represent a useful subset of possible systems as a matrix of numbers, and manipulation of the rows and columns of the matrix is identical to manipulating the longhand equations. This turns out to be astoundingly powerful. In computer based systems this is probably the single most common and powerful way of representing and solving problems. (I used to estimate that about half of all numeric scientific problems were linear algebra.) Any book on writing computer games will cover this material. What I’ve written above is really to encourage you to look into the question, and to have some idea of what you will meet.
