Modified 4PM Friday, Frb. 1 2002<\/p>\n
The magnitude of the drag force acting on the ball is FD<\/span><\/sub> = -k vw<\/sub>2<\/sup><\/span>. Then the acceleration in the x<\/i>-direction satisfies dvx<\/span><\/sub><\/i> \/ dt = [(-k\/m) ( vx<\/span><\/sub> – tailwind) sqrt( ( vx<\/span><\/sub> – tailwind)2<\/span><\/sup> + vy<\/sub>2<\/sup><\/span> ) ] g = 32 ft\/s\/s, k\/m = 32\/1402<\/span><\/sup>, at sea level. (remember, that k\/m will decrease at higher altitudes)<\/p>\n Thus, the Maple<\/i> code will look something like v0:= 180;
\nThe velocity relative to the ground is v<\/b> = vx<\/span><\/sub> i<\/b> + vy<\/span><\/sub> j<\/b>, so the velocity relative to the wind is v<\/b>w<\/span><\/sub> = ( vx<\/span><\/sub> – tailwind) i<\/b> + vy<\/span><\/sub> j<\/b>
\nThe speed relative to the wind is then vw<\/span><\/sub> = sqrt( ( vx<\/span><\/sub> – tailwind)2<\/span><\/sup> + vy<\/sub>2<\/sup><\/span> ).
\nThe direction, relative to the wind is u<\/b>w<\/span><\/sub> = [( vx<\/span><\/sub> – tailwind)\/vw<\/span><\/sub>] i<\/b> + [ vy<\/span><\/sub>\/vw<\/span><\/sub>] j<\/b>.
\nThe force vector is parallel to v<\/b>w<\/span><\/sub>, so F<\/b>D<\/span><\/sub> = FD<\/span><\/sub> u<\/b>w<\/span><\/sub> and
\nF<\/b>D<\/span><\/sub> =-k vw<\/sub>2<\/sup><\/span> { [( vx<\/span><\/sub> – tailwind)\/vw<\/span><\/sub>] i<\/b> + [ vy<\/span><\/sub>\/vw<\/span><\/sub>] j<\/b> }.
\nTherefore, F<\/b>D<\/span><\/sub> = [-k ( vx<\/span><\/sub> – tailwind)vw<\/span><\/sub>] i<\/b> + [-k vy<\/span><\/sub> vw<\/span><\/sub>] j<\/b>.<\/p>\n
\nThe acceleration in the y<\/i>-direction satisfies dvy<\/span><\/sub><\/i> \/ dt = -g – [(k\/m) v sqrt( ( vx<\/span><\/sub> – tailwind)2<\/span><\/sup> + vy<\/sub>2<\/sup><\/span> ) ]<\/p>\n
\n<\/tt><\/p>\n
\ntheta:= .9;
\ntailwind:=10;
\nfly:= dsolve(
\n{ diff(x(t),t,t) = -32\/140^2 * sqrt( (diff(x(t),t)-tailwind)^2 + diff(y(t),t)^2)* (diff(x(t),t)-tailwind) ,
\ndiff(xyt),t,t) = -32 -32\/140^2 * sqrt( (diff(x(t),t)-tailwind)^2 + diff(y(t),t)^2) * diff(y(t),t) ,
\nx(0)=0, y(0)=0, D(x)(0)=v0*cos(theta), D(y)(0)=v0*sin(theta) },
\n{x(t),y(t)}, numeric, output=listprocedure );
\nxt:= subs( fly, x(t)) ;
\nyt:= subs( fly, y(t));
\nvxt:= subs( fly, diff(x(t),t));
\nvyt:= subs( fly, diff(y(t),t));<\/p>\n
\n<\/span><\/tt> <\/span><\/p>\n