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MathGroup Archive 1999

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Re: Mathematica can't win against Tiger Woods

  • To: mathgroup at smc.vnet.net
  • Subject: [mg19687] Re: Mathematica can't win against Tiger Woods
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Sat, 11 Sep 1999 16:35:58 -0400
  • Organization: Universitaet Leipzig
  • References: <7r7jvo$ck4@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi William,

first of all the result is not very complicated. It is less than 10
printed pages and it has no special function like a hypergeometric one
in it.

Typical a computer algebra uses the most general result or the most
general algorithm. It must do so to fit into all possible cases.
Since Mathematica knows nothing about your problem it can't know that
g is real and positive. 

For your problem it seems to be better to calculate the velocities x'[t]
and y'[t]
and get
{{vx[t] -> (b*g)/(a^2 + b^2) + 
    (C[1]*Cos[b*t] - C[2]*Sin[b*t])/E^(a*t), 
  vy[t] -> -((a*g)/(a^2 + b^2)) + 
    (C[2]*Cos[b*t] + C[1]*Sin[b*t])/E^(a*t)}} 

a very simple one. It is the second integration (to obtain x[t] and
y[t])
that causes Mathematica to split Sin[] and Cos[] int exponentials and it
must do so due to the Exp[-a*t] term.  
If you integrate the vx[t] and vy[t] to obtain the positions and run
FullSimplify[] again you get

{(E^(a*t1)*(b*g*t1 + a*C[1] - b*C[2]) + 
    (-(a*C[1]) + b*C[2])*Cos[b*t1] + (b*C[1] + a*C[2])*
     Sin[b*t1])/((a^2 + b^2)*E^(a*t1)), 
  (E^(a*t1)*(-(a*g*t1) + b*C[1] + a*C[2]) - 
    (b*C[1] + a*C[2])*Cos[b*t1] + (-(a*C[1]) + b*C[2])*
     Sin[b*t1])/((a^2 + b^2)*E^(a*t1))}

and the answer can't be more more simple.

Hope that helps
  Jens



"William M. MacDonald" wrote:
> 
> I want to use the study of golf drives in teaching theoretical methods.  An
> approximate pair of equations to get insight assumes that the drag force is
>  linearly  proportional to velocity, instead of the actual quadratic
>  dependence.  The equations for a ball with backspin to provide lift are
>      x''[t]== - (a x'[t]+b y'[t]),
>      y''[t]== - g - (a y'[t]- b x'[t])
>  Mathematica returns a very complicated and apparently complex expression in
>  about 9 seconds on my 250 MHz G3 Powerbook.  Simplify takes 1min and 20
>  seconds and still returns an apparently complex expression.  If I apply
>  FullSimplify on the solution for say x[t], I get no answer in 6 minutes.
> 
>      I have a PC version of another system that I can run on my Powerbook using
>  Virtual PC.  It requires 6 seconds to deliver a lengthy but obviously real,
>  no Exp[(a+ I b)t] terms or (a + I b)(a - I b) terms.
> 
>      I have never been able to learn why Mathematica is so slow in solving
>  coupled equations and returns (as USUAL unless you use Simplify) such
>  inelegant results.  Is there any computer algebra NERD out
>  there who knows the answer.  (Don't tell me to use AlgebraicManipulation; I
>  am trying to sell Mathematica to users who don't want to spend time
> learning
>  fancy tricks.)
> 
> --
> William M. MacDonald
> Professor of Physics
> University of Maryland
> 
> Internet: wm2 at umail.umd.edu


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