# Manipulate showing the trajectory of a particle

I'm trying to plot the trajectory of a particle:

Vx = 741.61;
ay = 9.81;

Sx[t_] = Vx * t;
Sy[t_] = 1.5 - 1/2 * ay * t^2;
Vy[t_] = -ay * t;
Sy2[x_] = Sy[x/Vx];
Manipulate[
Show[
Graphics[ Arrow[{{Sx[t], Sy[t]}, {Sx[t] + Vx/10, Sy[t]}}],
Axes -> True, PlotRange -> {{0, 500}, {-1, 2}},
AspectRatio -> Full, ImageSize -> {500, 300}],
Graphics[Arrow[{{Sx[t], Sy[t]}, {Sx[t], Sy[t] + Vy[t]/10}}],
Axes -> True, PlotRange -> {{0, 500}, {-1, 2}},
AspectRatio -> Full, ImageSize -> {500, 300}],
Plot[Sy2[x], {x, 0, Sx[0.553]}]
]
, {t, 0, 0.553}] I would like the plot to be drawn as the projectile flies, but when I replace Plot[Sy2[x], {x, 0, Sx[0.553]}] with Plot[Sy2[x], {x, 0, Sx[t]}] Mathematica complains.

How can I change the range of the plot with Manipulate?

I would like the plot to be drawn as the projectile flies

You mean something like this? You had few errors. Functions need to be defined using := and not =, and for initial setting, used 0.01 in Plot[sy2[x], {x, 0.01, sx[t]}] to prevent range from being zero at start, else M will complain. And few other minor issues.

Manipulate[
Show[

Graphics[Arrow[{{sx[t], sy[t]}, {sx[t] + vx/10, sy[t]}}],
Axes -> True,
PlotRange -> {{0, 500}, {-1, 2}},
AspectRatio -> Full,
ImageSize -> {500, 300}],

Graphics[Arrow[{{sx[t], sy[t]}, {sx[t], sy[t] + vy[t]/10}}],
Axes -> True,
PlotRange -> {{0, 500}, {-1, 2}},
AspectRatio -> Full, ImageSize -> {500, 300}],

Plot[sy2[x], {x, 0.01, sx[t]}]
],
{{t, 0, "time"}, 0, maxT, .01, Appearance -> "Labeled"},
Initialization :>
(
maxT = 0.553;
vx = 741.61;
ay = 9.81;
sx[t_] := vx*t;
sy[t_] := 1.5 - 1/2*ay*t^2;
vy[t_] := -ay*t;
sy2[x_] := sy[x/vx]
)
]


Just for variety, you can exploit vectorization.

Setup:

s[vx_, vy_, t_, s0_] := {vx t, s0 + vy t - 9.81 t^2/2}
v[vx_, vy_, t_] := {{vx, 0}, {0, vy - 9.81 t}};
gnd = t /. First@Solve[Last@s[746.1, 0, t, 1.5] == 0 && t > 0, t];


Visualization:

pp[m_] :=
With[{b = s[746.1, 0, m, 1.5], vel = v[746.1, 0, m]},
ParametricPlot[Evaluate[s[746.1, 0, t, 1.5]], {t, 0, m},
Epilog -> ({Arrowheads[0.05], Arrow[{b, b + 0.1 #}]} & /@ vel),
PlotRange -> {{0, 600}, {0, 2}}, AspectRatio -> Full]]

Manipulate[pp[time], {time, 0.01, gnd}] 