Session 9: Week 17/18: Projectiles

The purpose of this exercise is to calculate the theoretical trajectory of a projectile in a constant gravitational field. In other words, we want to calculate the path followed by, say, a ball thrown in the air.

We are going to make some simplifications. We will work in just two dimensions (i.e. as if we are looking exactly side on at the path of the ball). We will also neglect air resistance, and will treat the projectile as a point mass. This allows a simple analysis of the motion of the projectile, using newtonian theory.

In our two dimensional ( and ) coordinate system, will denote horizontal position, and will denote vertical position.

The initial value of will be taken as zero.

A zero value will correspond to the projectile being on the ground. The initial value of will be some positive value (in effect, the height of the person throwing the ball).

will denote the horizontal component of velocity, and the vertical component. We will assume that both and are initially positive. Similarly, and will be the components of acceleration.

The "initial" information provided for a particular run of the program (by assigning explicit values to suitable variables) will be the initial height () and initial velocity components ( and ) of the projectile. These values will be assigned to suitable variables. You can either code values directly into the program (in which case you will have to change the program, and rebuild, every time you want to try different values) or you can have the program prompt for and read in values when it starts running.

The output from the program will be a print out on the screen of values of and at successive, "small", intervals of time. The program should terminate when becomes zero again (i.e. the projectile falls back to earth). This print out may be captured into a text file (see the previous discussion of redirection) and a graph plotted of versus (one way of doing this is described in detail in the Hints section below).

Informally, the motion of the projectile is pretty simple: it moves steadily to the right (positive direction), simultaneously going up for a while (positive direction), and then coming down again to land.

Formally, at each time step, the new co-ordinate can be calculated as:

where (approximately):

In words, if the velocity in the direction is , then, in a time , the projectile will move a distance of multipled by ; and the new position is the old position plus this movement. This calculation would be exact if were constant; if is changing (as it will be in our case) then the calculation will still be approximately correct, as long as as is "small" - namely small enough that the percentage change in in that time is "negligable". Of course, you can only estimate this once you have some idea of how quickly will be changing...

The new co-ordinate may be calculated similarly.

Now, in principle, both and could also be simultaneously changing, and have to be recalculated, at each time step. But since we are neglecting air resistance, and since gravity works straight down, there will be zero acceleration in the direction ( is zero), so that will actually remain constant, equal to its initial value. There will be acceleration in the direction however, of value , where is the acceleration due to gravity at the surface of the earth (about 9.81 meters per second per second). This is negative because our co-ordinate is positive in the upward direction and gravity works downward (last time I checked anyway...). This acceleration in the direction will be constant (since is not changing). Thus, the new at each time step can be calculated as follows:

where:

That is, the change in the (vertical) component of speed is simply the (vertical) acceleration multiplied by the time it acts over; and the new speed is simply the old speed plus this change.

You should test your program carefully. Explain, in your report, what tests you carry out, how the program behaves, and whether it has passed such tests.

In the first place, the repetition or iteration in your program should be achieved using a while statement. However, if you succeed in getting this working satisfactorily, you should then experiment with using a for statement instead.

• Hints