(* ------------------------------------------------------------- *)(*   a618-curve-family.nb  14 February 2013   [Richard Hall]     *)
(* ------------------------------------------------------------- *)

(*    a618 Q(1.4)     u = g = 1    *)

x[t_,a_] := Cos[a] t

y[t_,a_] :=- t^2/2 + Sin[a]t

(* One curve for  a = 1    *)

ParametricPlot[{x[t,1],y[t,1]},{t,0,Pi},PlotRange->{{0,1},{0,1/2}}]

(* generate the pair of functions for a given  a    *)

xy[a_] := {x[t,a],y[t,a]}

ParametricPlot[xy[1],{t,0,Pi},PlotRange->{{0,1},{0,1/2}}]

(* The Map function  *)

Map[Sin,{1,2,3}]

(* a useful mesh function: n steps  n+1 values in [a,b] *)

mesh[a_,b_,n_] := Table[a+ (i-1)*(b-a)/n,{i,n+1}]

mesh[1,2,5]

(* Plot curves for 11 values of a in [0.2,1.5]   *)

ParametricPlot[Map[xy,mesh[0.2,1.5,10]],{t,0,Pi},PlotRange->{{0,1},{0,1/2}}]