(1) Kingdom of Catapults is related to the mathematical theory of projectiles. This theory is concerned with the motion of an object that is launched into the air.
(2) In Kingdom of Catapults you have control of the launch angle and launch speed of projectile pieces of fruit. Altering the speed and angle affect both the path of the projectile, and the time it takes for the projectile to land. The fruits are of different weights, so some can be launched with greater initial speeds than others.
(3) We ignore air resistance in Kingdom of Catapults, which means that the initial launch angle and speed determine the motion of the projectile completely. The distance travelled by the projectile can be calculated from an equation.
(4) The path followed by a projectile is known as its trajectory. Without air resistance the trajectory has the shape of a parabola. A parabola is a mathematical curve with many important properties. For example, the mirrors in car headlights have a parabolic shape in order to focus light in a beam.
(5) Remarkably, without air resistance, the trajectory and time of flight of a projectile do not depend on the mass of the projectile. A cannon ball will travel the same distance and for the same length of time as a marble.
(6) The trajectory of a projectile is also affected by gravity. Because the Sun has greater mass than the Earth, a projectile fired from the Earth would travel farther than the same projectile fired from the Sun. On the other hand, a projectile fired from the Moon would travel farther than the same projectile fired from the Earth.
(7) When you fire a projectile vertically in the air the projectile is, for an instant, motionless when it reaches its highest point. The projectile is never motionless when the launch angle is not vertical; however, at the highest point of its parabolic trajectory its vertical speed is zero, but its horizontal speed is not zero.
(8) In fact, without air resistance there are no horizontal forces acting on the trajectory, and its horizontal speed remains constant throughout its motion.
(9) When firing from flat ground, and ignoring air resistance, the maximum range of a projectile is achieved when you fire at a 45 degree angle to the horizontal ground. In Kingdom of Catapults you fire from a high castle, and the maximum range is achieved when you fire at an angle less than 45 degrees to the horizontal. (The precise angle depends on the height of the castle.)
(10) The equations of motion of a projectile are more complicated when air resistance is considered. The size and shape of the projectile both affect air resistance. The trajectory is no longer typically a parabola.
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Trajectories
I share the other writers disappointment with The Code and with the material associated with the catapult.
The confusion which some people have in relation to these matters results from their experience of how balls etc move and their inability to separate these from the what The Code seems to see as the arcane world that only pure mathematics can reveal. It reminds one of the ancient 'gnostic' idea of secret knowledge that only the initiated can know and reveal to the rest of us.
Engineering science has dealt with the trajectory problem in the real world and provides solutions which are at once both fascinating and useful - which is more than can be said for the catapult problem.
This program for predicting trajectories together with a User Manual is freely available at http://repository.unimelb.edu.au/10187/440
An associated monograph by the writer -The Mechanics of Fluid - Particle System' is also available at bookshops and at http://www.bookshop.unimelb.edu.au/cbc/p?9781921775178
The theory which appears in the monograph will clarify the influence of air resistance on real bodies from water droplets to sports balls; the zero air resistance condition can also be studied.
I understand the difficulties of presenting such a topic to an audience with a wide range of backgrounds. However that understanding is not helped by setting the topic in a context where, notwithstanding many interesting commonalities, it is claimed or at least suggested there is a single 'scientistic' code which somehow determines the way the world works.
Statement 5 is Incorrect.
"(5) Remarkably, without air resistance, the trajectory and time of flight of a projectile do not depend on the mass of the projectile. A cannon ball will travel the same distance and for the same length of time as a marble. In reality, a cannon ball would feel more air resistance (and the marble would travel farther, provided that it had the same density as the cannon ball)."
The cannon ball would indeed feel more air resistance.
However it is also proportionally much more massive than the marble.
For objects of identical shape and density, air resistance scales by the square of size, and mass the by cube of size.
And as acceleration is force over mass.
The deceleration of the larger cannonball would be less than the deceleration of the marble.
Thus the marble will go farther.
On the topic of the BBC/OU programs themselves.
I was disappointed by the total lack of depth, yet again the program was aimed entirely at those with absolutely no knowledge of mathematics or science with nothing to interest those who are interested in the subject and have some knowledge of it.
Also the format with the random whispering voices uttering fragments of phrases from the program did nothing but irritate and waste air time that could have been used for greater depth.
Thanks for pointing out my
Thanks for pointing out my mistake, David. I will arrange for the final part of statement five to be removed. I agree with what you say (apart from your final statement, which should be "thus the cannon ball will go farther") but I'd rather avoid those details in my comments.
As for the TV series, yes, it was meant to be accessible to an audience with limited mathematical background. I thought that there were features of interest to experienced mathematicians, although, as you indicate, these features weren't explored in detail. To be shown on mainstream BBC 2 (rather than more specialised BBC 4) the series had to be understood by a large audience.
Thanks again,
Ian
Correcting the correction
Thank you for responding,
I was expecting at most a moderator response, and was pleased and
surprised to get one from you.
oops, yes I definitely meant the cannon ball would go farther.
I understand your reasoning about pitching the level of the program.
And it is a tricky issue to get right.
However I still don't agree that, in this instance, the collection of people
responsible for this decision got it right.
Certainly if the irritating 'whispering voices' had been removed you
could have spent a lot more time explaining maths rather than trying to
sound mysterious.
The point of science and maths is to illuminate the world and explain things
not make things look mysterious for the sake of effect.
That's the job of mystics and the religious.
One example from the program that springs to mind is your treatment
of the object i.
You said that i is a mysterious number with the weird property that i*i=-1
breaking the rule that a number times itself is always a positive.
You then went on to say that this weird entity is very useful, which it is.
But, you never gave any explanation for what i is and why it has this weird
property, you just conjured it from thin air, like a magician pulling a rabbit
from a hat.
i is of course sqrt(-1), and it is easy to see why it is written as i as you can't
represent sqrt(-1) as a real number.
This also makes it clear why i*i=-1,
sqrt(a)*sqrt(a)=a... in this case a is -1 thus meaning that i*i=-1.
Suddenly this thing i has a reason for existing and an explanation for the strange
property you highlighted and all you needed was pre gcse maths.
When trying to pitch these programs you should always have something in there
to push back on the boundaries of what the target audience can grasp.
Not all of them will get it, but some will.
The frustration I have is that everything gets pitched down to the lowest common
denominator, or lower.
Also, I don't think its good to regard the audience for the BBC's two flagship channels
as being dumber or more ignorant than those with digital on BBC 4.
If you are truly trying to inspire people to go further, then put in a proper explanation
for the effects of air resistance.
Take the next step rather than being permanently stuck on the first.
Again, Thank you for responding to my post.
David Heath
Thanks for these reasonable
Thanks for these reasonable comments. I'm not in a position to offer a constructive response though, as I was only a consultant for the series. You mention the segment on i: yes, there wasn't much detail - sorry that you found it frustrating. Regarding the BBC2/BBC4 split, my understanding is that BBC2 programmes must be accessible to as large an audience as possible. I emphasise that's only my understanding of the BBC policy though (and perhaps in making this point I am just repeating myself).
Finally, on the ten statements about projectiles on this page, they are meant to be tips for playing the Flash game. Ideally I would accompany the tips with a link to another page about the mathematics behind projectiles. Then I could include a brief account of air resistance. I haven't the time for this though!
If the BBC receive feedback asking for higher level mathematics programmes, then perhaps they will oblige. Certainly such programmes would have my support.
Thanks again for your input.
Ian