Fw: balls

["Don Williamson" <DonWilliamson@aqua-pacific.com>]

Matt, here's the original post. And Andy, I was taught all of this and a
hell of a lot more (like how to model a mesh of rods falling from the sky
and colliding with various pegs, other rods and even rods suspended from
springs) at school... you just gotta pick the right classes :0)

- Don

-----Original Message-----
From: Don Williamson <donwilliamson@aqua-pacific.com>
To: gameprogrammer@gameprogrammer.com <gameprogrammer@gameprogrammer.com>
Date: 03 September 1999 10:29
Subject: Re: balls


>Hallo,
>
>here's a perfect example of "applying what you once learned in a maths
>lesson". It's just a simple case of Conservation of Momentum and Newtons
>Experimental Law. It's a bit difficult to explain without diagrams but here
>goes (explained with 1D).
>
>When the balls bounce around they'll either hit each other, or hit a solid
>wall I assume. Before a ball goes into collision it will have velocity
"u1".
>When it leaves the collision it can have a different velocity "v1". If it
is
>colliding with another ball then the same can be said for this one, the
>velocity going into the collision (u2) can be different from the velocity
>going out of the collision (v2). The important thing to note is that
>momentum is conserved throughout this collision, a simple transference
>occurs. Momentum is calculated like...
>
>    momentum = velocity * mass
>
>And Conservation of Momentum states that the total momentum involved before
>the collision is exactly the same as that total momentum after the
>collision...
>
>    m1 * u1 - m2 * u2 = m1 * v1 + m2 * v2
>
>The reason I'm doing a subtract here is because both balls are travelling
>towards each other, hence in different directions. m1 and m2 are the masses
>of your two balls. You should know the velocity going into the collision
and
>you know the masses so immediately a big chunk of the equation becomes easy
>to calculate. What you need now is the velocities of the balls coming out
of
>the collision (v1 and v2) but this is impossible with the current equation
>since there are two unknowns. You now have to resort to something called
>Newtons Experimental Law which is simply...
>
>    v1 - v2 = -e(u1 - u2)
>
>The constant "e" in this equation is called the Co-efficient of
Restitution.
>Basically, the bigger this value, the more bouncy your collision will be.
>This value can vary like 0 < e < 1. (I don't believe in e=1 :0). e is a
>value which you can make up yourself and modify but it is a constant that
>varies depending on the types of both balls, not just one of them.
>
>So you know u1, u2 and e, what you have here is a simple simultaneous
>equation! (remember THEM from maths? :0)
>
>Let's say I had the following values...
>
>    e = 0.2
>    u1 = 2
>    u2 = -3            (moving the other way)
>    m1 = 1
>    m2 = 2
>
>I would now go onto calculate the collision as follows...
>
>    Conservation of Momentum:
>    1 * 2 - 2 * 3 = v1 + 2 * v2
>    2 - 6 = v1 + 2 * v2
>    -4 = v1 + 2 * v2
>
>    NEL:
>    v1 - v2 = -0.2(2 + 3)
>    v1 - v2 = -0.2(5)
>    v1 - v2 = -1
>
>    Simultaneous equations are:
>    v1 + 2(v2) = -4
>    v1 - v2 = -1
>
>    (multiply second equation by 2)
>
>    2(v1) - 2(v2) = -2
>
>    (add the two equations together to get rid of v2)
>
>    3(v1) = -6
>    v1 = -2
>
>    (substitute v1 into any equation to calculate v2 - 1st for example)
>
>    -2 + 2(v2) = -4
>    2(v2) = -2
>    v2 = -1
>
>This is great, whenever I write examples like this the numbers I choose
>always provide nice answers - weird :0)
>
>But that's the gist of it. If you want to extend the idea into 2D or 3D
>space then it's really simple, just remember that you MUST resolve your
>velocities along the x, y and z axes.
>
>I noticed that you have a UK mail account so maybe you're from the UK :0).
>If so then try and take Maths and Further Maths for A-Levels. If you do so
>then making calculations like this will become second nature and will
become
>a lot more fun. Hell, you'll even be able to apply collisions to these
>babies when they're spinning, ain't that cool :0)
>
>Seeya,
>- Don
>
>-----Original Message-----
>From: Eudoxus3 <Eudoxus@freeuk.com>
>To: Game programmer <gameprogrammer@gameprogrammer.com>
>Date: 03 September 1999 09:37
>Subject: balls
>
>
>>Hi,
>>
>>Does anyone have any routines for calc'ing many bouncing ball's movements.
>>
>>
>>Regards,
>>
>>Matt. W.
>>--
>>***  eudoxus@freeuk.com  ***  http://members.xoom.com/eudoxusM/
>>                 mwebster@apsoft.co.uk
>>
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