This is a program to practice shooting a ball into a ring on the floor.

(click to see video)

The problem is to figure out the angle to shoot a ball to get it into a ring on the floor. (The ball doesn't bounce or roll. You can think of the ring as being filled with sand.)

The exercise is to write a procedure targetBallShooter() that returns angle to shoot a ball. The procedure is given the x position of the ball shooter, the ball's speed (once shot), the x position of the ring, and the acceleration due to gravity. The procedure should computer the angle to shoot the ball, and return the value.

The procedure has access to the global state of the game, but it is not allowed to change any. These state variables are:

• ballShooter_x0 is the x position of the ball shooter
• ring_x0 is the x position of the ring
• ballShooter_speed is the speed with which a ball is shot.
• gravity is the gravitational acceleration on the ball.

In this section I'm going to give a walk-thru in the steps of solving the problem (and generally how to set of the software to meet challenges in games like this.)

This walk-thru is targeted to an advanced middle school level. This problem doesn't rely heavily on deriving the equations -- but you're expected to work out how to get them into the computer program appropriately. At an advanced high-school level, the student should derive the equations.

I will use diagrams; it's tempting to dismiss them, as they are "just" pictures. Do not. They help break the problem down, make it easily solvable (or at least, easier to solve), organize the information you need, and help reduce mistakes.

The first step is diagram the basic event that is happening: the who's, the when's, and the where's. The objective is to shoot a ball into the ring on the floor. So the "who's" are a ball and a ring. The "where" is the place they hit each other. And the "when" is the point in time the hit each other:


The next step is to describe the ball and it's flight parameters in a "parameter diagram":


This describes the balls straight-line flight at angle from the point it was shot. (We'll add gravity in next) The input to modelling the ball is it's time in the flight, and the result is the location after it's flown.

Now that we've described the basic ball, the next step is describe the forces on the ball, using a "free-body diagram":


This diagram is used to help work out all of the forces acting on the ball. The first step is to draw the actions by the ball, and then fill in with the forces that act on the ball. (In later projects we'll see how some of the forces on the ball are done in response to the action by the ball). The new forces will be added to the parameter diagram in the next step.

Once the free-body diagram has been fleshed-out, the next step is to update the parameter diagram with the added forces and how they affect the ball's behaviour:


The gravity force was added to the diagram in a lower box (for things that act on an object all the time, or very quickly). A second box was added to capture what the gravity does to the ideal motion of the ball, and changes that into a model of an "actual" motion. As this is targeted toward (advanced) middle school students, the formulation for motion is given. (Advanced) high school students should derive the formula.

The next step is to solve the equation. For the (advanced) middle school student, I will walk thru the steps to this using the C-language. There are many other ways of solving this. The (advanced) high school student should work thru the equations and solve them in a closed-form; that is, should come up with equations that do not use iteration.

First, you need to come up with an equation that tells you how long a ball will fly, before it hits the floor. To do this, you will need apply the Quadratic Equation to the formula. You will get two equations: one will give negative time values, and the other will give positive time values. Use the one that gives positive time values.

Next, we'll use a brute force technique to finding a suitable angle. Use a 'for' loop with an angle variable going from 0 degrees to 90 degrees. Use steps of 1 degree. The body of the for loop is:

1. Calculate the time that the ball will hit the floor. Use the new equation you came up with earlier.
2. Calculate where the ball travelled to and landed using the formula for the "x" variable, the time and angle.
3. Calculate how close this landing is with the x location of the ring (ring_x0)
4. Is this closer than the other attempts? If so, save the angle.

When done iterating, return the best angle.

I'm not going to give explicit steps how to make this algebra, nor am I going to give explicit steps on how to code it. That's your job.

The was created using the Xcode editor running under Mac OS X 10.8.x or later.