Drag Coefficient

Specification of the quadratic model is complete once the drag coefficient is implemented. From this information it is possible the smoothness of the solution can be determined.

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> kk := 1/2 * rho * piecewise( t<=t0, 1.95*b0,

> t<=t1, 1.95*b0 + 0.35*b1*l*((t-t0)/(t1-t0))^p,

> t<=t2, 0.35*b1*h + 1.33*Ae12,

> t<=t3, 0.35*b1*h + 1.33*Ae23,

> t> t3, 0.35*b1*h + 1.33*a1 );

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With the parameter values suggested in the text, the drag coefficient becomes completely well-defined

> K := simplify( subs( param, kk ) );

> dK := diff( K, t );

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Notice that the drag coefficient during final descent is approximately 29. This value is a little lower than eiter prediction based on the terminal velocity. This suggests that the landing velocity is likely to be a little higher than that for a jump from a 5-foot high wall and probably noticeably higher than the rate of descent published in [ebk, p. 86]. The precise amount of this over-estimate remains to be seen. The drag coefficient during free-fall is much closer to the value obtained from the literature.

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The graphs of and its derivative provide a first test of the smoothness of this function. However, be very careful before using a picture as a proof of continuity or smoothness.

> plotK := plot( K, t=0..30, discont=true, color=RED, axes=FRAMED, title="k(t) vs. t ( as given in (3.5) )" ):

> display( plotK );

> plotdK := plot( dK, t=0..30, discont=true, color=BLUE, axes=FRAMED, title="k'(t) vs. t" ):

> display( plotdK );

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A more reliable method of determining continuity of a function at a point is to compare its right- and left-hand limits at the point, i.e., the jump of the function at the point. The jump command defined at the top of this worksheet is designed for exactly this purpose. (This might be a nice place to use a Maple spreadsheet in Release 5.1.)

> for T in eval([t0,t1,t2,t3],param) do

> sprintf( "[k(%5.2f)] = %14.7f, [k'(%5.2f)] = %14.7f", T, eval( jump(K,t=T) ), T, eval( jump(dK,t=T) ) );

> od;

Thus, the drag coefficient is continuous but is not differentiable. This means the acceleration will be continuous, but not the jerk.

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