One-dimensional motion

Animated graphic demonsrating one-dimensional particle motion (in class, I can “play” this animated graphic to make the large dot move back and forth along the thick line segment; here the position of the dot at time t is given by p(t) = 2t^2 - 4t - 1, corresponding to the handout).

p[t_] = 2t^2 - 4t - 1;
q[t_] = 0;
Manipulate[ParametricPlot[{{p[s],q[s]}, {p[t] + .125 Cos[s],q[t] + .125 Sin[s]}}, {s, 0,4}, PlotStyle -> {{Thick,Blue},{Black, Thickness[.015]}},  PlotRange -> {{-5,17},{-1,1}}, Ticks -> {Range[-20,20],{}}]
, {{t,0}, 0, 4}]

An animated graphic showing position and velocity for this one-dimensional example (this shows the velocity of the dot at any time t as a vector (light-colored in this document, presently starting at the dot and ending at x = 8)

p[t_] = 2t^2 - 4t - 1;
q[t_] = 0;
Manipulate[ParametricPlot[{{p[s],q[s]}, {p[t] + .125 Cos[s],q[t] + .125 Sin[s]},
{p[t] + (s/4) p'[t], q[t] + .25}}, {s, 0,4}, PlotStyle -> {{Thick,Blue},{Black, Thickness[.015]}, {Orange}},  PlotRange -> {{-5,17},{-1,1}}, Ticks -> {Range[-20,20],{}}]
, {{t,0}, 0, 4}]

An animated graphic showing position, velocity and acceleration for this one-dimensional example:

p[t_] = 2t^2 - 4t - 1;
q[t_] = 0;
Manipulate[ParametricPlot[{{p[s],q[s]}, {p[t] + .125 Cos[s], q[t] + .125 Sin[s]},
{p[t] + (s/4) p'[t], q[t] + .25}, {p[t] + s, q[t] + .5}}, {s, 0,4}, PlotStyle -> {{Thick,Blue},{Black, Thickness[.015]}, {Orange}, {Red}},  PlotRange ->  {{-5,17},{-1,1}}, Ticks -> {Range[-20,20],{}}]
, {{t,0}, 0, 4}]

Two-dimensional motion

This animated graphic illustrates an example from my lecture notes, explaining two-dimensional motion in the plane (here the object’s position at time t is (t^2-1,t-t^2)).  In class, I play this animated graphic to show the motion of the dot along the path.

f[t_] = t^2-1;
g[t_] = t-t^2;
Manipulate[ParametricPlot[{{f[s],g[s]}, {f[t] + .125 Cos[s], g[t] + .125 Sin[s]}}, {s, -20, 20}, PlotStyle -> {Blue,{Black, Thickness[.01]}},  PlotRange -> {{-10,10},{-10,10}}]
, {{t,-10}, -10, 10}]

The same example from the notes, with velocity (I can animate this graphic to show how the velocity vector remains tangent to the object’s motion at any time)

f[t_] = t^2-1;
g[t_] = t-t^2;
Manipulate[ParametricPlot[{{f[s],g[s]}, {f[t] + .125 Cos[s], g[t] + .125 Sin[s]},
{f[t] + (s+40)/80 f'[t], g[t] + (s+40)/80 g'[t]}}, {s, -40, 40}, PlotStyle -> {Blue,{Black, Thickness[.01]}, Orange},  PlotRange -> {{-10,10},{-10,10}}]
, {{t,-10}, -10, 10}]

Motion of a particle moving along a circle of radius 4 (another example I show in class)

bb[t_] = 4 Cos[t];
bc[t_] = 4 Sin[t];
Manipulate[ParametricPlot[{{bb[s],bc[s]}, {bb[t] + .125 Cos[s], bc[t] + .125 Sin[s]}}, {s, -20, 20}, PlotStyle -> {Blue,{Black, Thickness[.01]}},  PlotRange -> {{-10,10},{-10,10}}]
, {{t,-4 Pi}, -4 Pi, 4 Pi}]

“Warped circle” (a more interesting set of parametric equations)

mm[t_] = 4 Cos[t] - 2 Sin[3t];
nn[t_] = 4 Sin[t] - 2 Sin[5t];
Manipulate[ParametricPlot[{{mm[s],nn[s]}, {mm[t] + .125 Cos[s], nn[t] + .125 Sin[s]}}, {s, -20, 20}, PlotStyle -> {Blue,{Black, Thickness[.01]}},  PlotRange -> {{-10,10},{-10,10}}]
, {{t,-10}, -10, 10}]

A crazy shape (again, this can be animated to show the dot moving along the curve)

m[t_] = (1/8)t^2 + Cos[5t];
n[t_] = t +2  Sin[3t];
Manipulate[ParametricPlot[{{m[s],n[s]}, {m[t] + .125 Cos[s], n[t] + .125 Sin[s]}}, {s, -20, 20}, PlotStyle -> {Blue,{Black, Thickness[.01]}},  PlotRange -> {{-5,15},{-10,10}}]
, {{t,-15}, -15, 15}]

A figure-eight with the velocity vector shown (shows how the velocity vector changes with time)

a = 4 Pi;
b = 20;
h[t_] = 3 Cos[t];
k[t_] = 4 Sin[2 t];
Manipulate[ParametricPlot[{{h[s],k[s]}, {h[t] + .125 Cos[s], k[t] + .125 Sin[s]}, {h[t] + ((s+b)/(2b))h'[t], k[t]+ ((s+b)/(2b))k'[t]}}, {s, -b,b}, PlotStyle -> {Blue,{Black, Thickness[.01]}, {Orange}},  PlotRange -> {{-10,10},{-10,10}}]
, {{t,0}, -a, a}]

The figure-eight with velocity and acceleration indicated (shows the acceleration and velocity vectors; this graphic can be animated to show how the acceleration and velocity of the object impact its motion)

a = 4 Pi;
b = 20;
h[t_] = 3 Cos[t];
k[t_] = 4 Sin[2 t];
Manipulate[ParametricPlot[{{h[s],k[s]}, {h[t] + .125 Cos[s], k[t] + .125 Sin[s]}, {h[t] + ((s+b)/(2b))h'[t], k[t]+ ((s+b)/(2b))k'[t]},
{h[t] + ((s+b)/(2b))h''[t], k[t]+ ((s+b)/(2b))k''[t]}},
 {s, -b,b}, PlotStyle -> {Blue,{Black, Thickness[.01]}, {Orange}, Red},  PlotRange -> {{-10,10},{-10,10}}]
, {{t,-a}, -a, a}]