Parametric Equation Lab

Central Magnet School

Time: Menu

The student will

Graph Parametric Equations

Represent pojectile motion problems with parametric equations

Investigate collisions

parameters={a,c,d,x}

Parametric Basics

Parametric equations have two functions x(t) and y(t) where t is the variable in both. The easiest way to graph a parametric equation is to use substitution to convert the two equations into one function of y and x. To do that, simply solve for t in one of the equations and substitute into the other equation. For example,

({{({{x=3t
y= -6t+1}})}})

Therefore, t=(x/3). The substitute that in for t in the second equation and you should see that y=-6((x/3))+1. Thus, y=-2x+1 which can be easily graphed.

Plot[-2x+1,{x,-4,4}]

There are a couple of problems with graphing the function by eliminating t. Without t you can not see where the “particle” is at various times. Therefore, you can’t        tell the speed or direction of the movement. Lets examine a better way on slide 3.

Preserving t

In order to preserve t simply create a table of values and plug in seconds to t. For example,

({{({{x=-3t+10
y=2t-1}})}})

So,     ({{t, x, y}, {0, 10, -1}, {1, 7, 1}, {2, 4, 3}, {3, 1, 5}, {4, -2, 7}})

That is how it should look on paper. Now, lets bring it to life.

Sub[expr_,var_,val_]:=
expr /. var->val;
Sub[expr_,var_,vals_?ListQ]:=
Transpose[Sub[expr,var,#]& /@ vals];
ParametricTrace[{f1_,f2_},{var_,a_,b_},t0_,mode_,plotrange_,places_]:=
Module[{pl,dt,tx,gr,grobjs,t,x,y},

If[a<b,
pl= If[mode===dot || t0>a,First[ParametricPlot[{f1,f2},Evaluate[Switch[mode,dot,{var,a,b},trace,{var,a,t0},segment,{var,t0-0.1,t0+0.1}]],Axes->False,PlotStyle->Blue]],{}];
dt={PointSize[Medium],Style[Point[Sub[{f1,f2},var,t0]],RGBColor[1,.47,0]]};
grobjs = {pl,dt};
{t,x,y} = NumberForm[Chop[N[#,places+2]],{places+2,places},NumberPadding->{" ","0"},ExponentFunction-> (Null&) ]& /@  {t0,Sub[f1,var,t0],Sub[f2,var,t0]};
tx = Text@Row[{Style["t",Italic]," =",t,"  ",Style["x",Italic]," =",x,"  ",Style["y",Italic]," =",y}];

gr=Graphics[grobjs,Axes->True,AspectRatio->Automatic ,AxesOrigin->{Automatic,0}, ImageSize->{400,300},PlotRange->plotrange]
];

Grid[{{tx},{gr}}]
];
ParametricFunctionList = 
{{{-3t+10,2t-1},{{-4,10},{-4,10}},{0,4.72}}->"x = -3t, y = 2t-1"
};
Manipulate[If[t0>Last[Last[f]],t0=Last[Last[f]]];
ParametricTrace[First[f],Prepend[Last[f],t],t0,trace,First[Rest[f]],5],
{{f,{{-3t+10,2t-1},{{-4,10},{-4,10}},{0,4.72}},"equations"},ParametricFunctionList,BaseStyle->{10}},
{{t0,0,"t"},First[Last[f]],Last[Last[f]],Appearance->"Open"},
SaveDefinitions->True,AutorunSequencing-> {4}]

Graphing Parametric Equations

Graph each equation below from 0 to 5 seconds. Then check your graph with the demonstration below. Remember to add arrows to your graph to indicate the direction the particle is going.

1. {({{x, =, 4t}, {y, =, 2t}})                      2. {({{x, =, t-2}, {y, =, 4t}})                             3. {({{x, =, (t/4)}, {y, =, -3t}})                        4. {({{x, =, 20t}, {y, =, 10t+10}})

Sub[expr_,var_,val_]:=
expr /. var->val;
Sub[expr_,var_,vals_?ListQ]:=
Transpose[Sub[expr,var,#]& /@ vals];
ParametricTrace[{f1_,f2_},{var_,a_,b_},t0_,mode_,plotrange_,places_]:=
Module[{pl,dt,tx,gr,grobjs,t,x,y},

If[a<b,
pl= If[mode===dot || t0>a,First[ParametricPlot[{f1,f2},Evaluate[Switch[mode,dot,{var,a,b},trace,{var,a,t0},segment,{var,t0-0.1,t0+0.1}]],Axes->False,PlotStyle->Blue]],{}];
dt={PointSize[Medium],Style[Point[Sub[{f1,f2},var,t0]],RGBColor[1,.47,0]]};
grobjs = {pl,dt};
{t,x,y} = NumberForm[Chop[N[#,places+2]],{places+2,places},NumberPadding->{" ","0"},ExponentFunction-> (Null&) ]& /@  {t0,Sub[f1,var,t0],Sub[f2,var,t0]};
tx = Text@Row[{Style["t",Italic]," =",t,"  ",Style["x",Italic]," =",x,"  ",Style["y",Italic]," =",y}];

gr=Graphics[grobjs,Axes->True,AspectRatio->Automatic ,AxesOrigin->{Automatic,0}, ImageSize->{400,300},PlotRange->plotrange]
];

Grid[{{tx},{gr}}]
];
ParametricFunctionList = 
{{{4t,2t},{{-2,20},{-2,10}},{0,5}}->"x = 4t, y = 2t",
{{t-2,4t},{{-10,10},{-1,20}},{0,5}}->"x = t-2, y = 4t",
{{t/4,-3t},{{-10,10},{2,-15}},{0,5}}->"x = t/4, y = -3t",
{{20t,10t+10},{{-1,100},{-1,60}},{0,5}}->"x = 20t, y = 10t+10"
};
Manipulate[If[t0>Last[Last[f]],t0=Last[Last[f]]];
ParametricTrace[First[f],Prepend[Last[f],t],t0,trace,First[Rest[f]],5],
{{f,{{4t,2t},{{-2,20},{-2,10}},{0,5}},"equations"},ParametricFunctionList,BaseStyle->{10}},
{{t0,0,"t"},First[Last[f]],Last[Last[f]],Appearance->"Open"},
SaveDefinitions->True,AutorunSequencing-> {4}]

Projectile Motion

Projectile motion is the path by which object takes when tossed, thrown, or hit into the air. When gravity and an angle of elevation is involved we must use the following parametric equation where (Subscript[x, 0])is the initial horizontal position, (Subscript[y, 0])is the inital vertical position, (Subscript[v, 0])is the initial velocity, and θ is the initial angle of elevation.

x = (Subscript[x, 0])+(Subscript[v, 0])cosθ·t
                y= (Subscript[y, 0])+(Subscript[v, 0])sinθ·t-(1/2)(gt^2)

For example, a baseball is hit when it is 3 ft above the ground. It leaves the bat with a velocity of 150 ft/sec and at an angle of elevation of (20^o). A 20-foot high fence is 400 ft from home plate. Will it be a home run?

	Lets label what we know. ({{(Subscript[y, 0])= 3ft}, {                (Subscript[v, 0])=150ft/sec}, {    θ=(20^o)}})       with gravity (g) being 32 ft/sec. So our equation should be:

({{{({{x=150cos((20^o))·t}, {y=3+150sin((20^o))·t-16(t^2)}})}})

Watch the path of the ball below. After close inspection, it looks very close to 20 ft at 400ft away. In order to be exact, see what time the ball reaches 400 ft by solving 400=150cos(20)·t. Then plug that value of t into y(t) to see what the height is at that specific second. You should see that the height is 19.737ft at ≈ 2.83 seconds.

Sub[expr_,var_,val_]:=
expr /. var->val;
Sub[expr_,var_,vals_?ListQ]:=
Transpose[Sub[expr,var,#]& /@ vals];
ParametricTrace[{f1_,f2_},{var_,a_,b_},t0_,mode_,plotrange_,places_]:=
Module[{pl,dt,tx,gr,grobjs,t,x,y},

If[a<b,
pl= If[mode===dot || t0>a,First[ParametricPlot[{f1,f2},Evaluate[Switch[mode,dot,{var,a,b},trace,{var,a,t0},segment,{var,t0-0.1,t0+0.1}]],Axes->False,PlotStyle->Blue]],{}];
dt={PointSize[Medium],Style[Point[Sub[{f1,f2},var,t0]],RGBColor[1,.47,0]]};
grobjs = {pl,dt};
{t,x,y} = NumberForm[Chop[N[#,places+2]],{places+2,places},NumberPadding->{" ","0"},ExponentFunction-> (Null&) ]& /@  {t0,Sub[f1,var,t0],Sub[f2,var,t0]};
tx = Text@Row[{Style["t",Italic]," =",t,"  ",Style["x",Italic]," =",x,"  ",Style["y",Italic]," =",y}];

gr=Graphics[grobjs,Axes->True,AspectRatio->Automatic ,AxesOrigin->{Automatic,0}, ImageSize->{400,300},PlotRange->plotrange]
];

Grid[{{tx},{gr}}]
];
ParametricFunctionList = 
{{{140.95t,3+51.3t-16(t^2)},{{-2,550},{-2,100}},{0,3.26}}->"x =150cos((20^o))t, y = 3+150sin((20^o))t-16(t^2)"
};
Manipulate[If[t0>Last[Last[f]],t0=Last[Last[f]]];
ParametricTrace[First[f],Prepend[Last[f],t],t0,trace,First[Rest[f]],5],
{{f,{{140.95t,3+51.3t-16(t^2)},{{-2,550},{-2,100}},{0,3.26}},"equations"},ParametricFunctionList,BaseStyle->{10}},
{{t0,0,"t"},First[Last[f]],Last[Last[f]],Appearance->"Open"},
SaveDefinitions->True,AutorunSequencing-> {4}]

Golfing

A golfer hits a ball with an initial velocity of 133 ft/sec and at an angle of (36^o) from the horizontal. Find when and where the ball will hit the ground. Will the ball clear a fence 9 ft high that is at a distance of 275 ft from the golfer? Show your work.

Sub[expr_,var_,val_]:=
expr /. var->val;
Sub[expr_,var_,vals_?ListQ]:=
Transpose[Sub[expr,var,#]& /@ vals];
ParametricTrace[{f1_,f2_},{var_,a_,b_},t0_,mode_,plotrange_,places_]:=
Module[{pl,dt,tx,gr,grobjs,t,x,y},

If[a<b,
pl= If[mode===dot || t0>a,First[ParametricPlot[{f1,f2},Evaluate[Switch[mode,dot,{var,a,b},trace,{var,a,t0},segment,{var,t0-0.1,t0+0.1}]],Axes->False,PlotStyle->Blue]],{}];
dt={PointSize[Medium],Style[Point[Sub[{f1,f2},var,t0]],RGBColor[1,.47,0]]};
grobjs = {pl,dt};
{t,x,y} = NumberForm[Chop[N[#,places+2]],{places+2,places},NumberPadding->{" ","0"},ExponentFunction-> (Null&) ]& /@  {t0,Sub[f1,var,t0],Sub[f2,var,t0]};
tx = Text@Row[{Style["t",Italic]," =",t,"  ",Style["x",Italic]," =",x,"  ",Style["y",Italic]," =",y}];

gr=Graphics[grobjs,Axes->True,AspectRatio->Automatic ,AxesOrigin->{Automatic,0}, ImageSize->{400,300},PlotRange->plotrange]
];

Grid[{{tx},{gr}}]
];
ParametricFunctionList = 
{{{107.599t,78.175t-16(t^2)},{{-1,550},{-1,100}},{0,4.886}}->"x =133cos((36^o))t, y = 133sin((36^o))t-16(t^2)"
};
Manipulate[If[t0>Last[Last[f]],t0=Last[Last[f]]];
ParametricTrace[First[f],Prepend[Last[f],t],t0,trace,First[Rest[f]],5],
{{f,{{107.599t,78.175t-16(t^2)},{{-1,550},{-1,100}},{0,4.886}},"equations"},ParametricFunctionList,BaseStyle->{10}},
{{t0,0,"t"},First[Last[f]],Last[Last[f]]},
SaveDefinitions->True,AutorunSequencing-> {4}]

Darts

A dart is thrown upward with an initial velocity of 58 ft/sec at an angle of elevation of (41^o). Neglect air resistance.
		a. Find the parametric equations that model the problem.

b. Draw a complete graph of the model.

c. What portion of the graph represents the problem situation?

d. When will the dart hit the ground?

e. Find the maximum height of the dart. At what time will the dart reach this maximum height?

f. How far does the dart travel in the horizontal direction?

Collisions

Even though some parametric equations intersect, they may not collide. For example, lets take the equations {({{x=2t            }, {y=-3t+8}})  and {({{x=4t}, {y=3t}})  . Notice their paths intersect, but the “particles” do not collide.

Sub[expr_,var_,val_]:=
expr /. var->val;
Sub[expr_,var_,vals_?ListQ]:=
Transpose[Sub[expr,var,#]& /@ vals];
ParametricTrace[{f1_,f2_},{var_,a_,b_},t0_,mode_,plotrange_,places_]:=
Module[{pl,dt,tx,gr,grobjs,t,x,y},

If[a<b,
pl= If[mode===dot || t0>a,First[ParametricPlot[{f1,f2},Evaluate[Switch[mode,dot,{var,a,b},trace,{var,a,t0},segment,{var,t0-0.1,t0+0.1}]],Axes->False,PlotStyle->Blue]],{}];
dt={PointSize[Medium],Style[Point[Sub[{f1,f2},var,t0]],RGBColor[1,.47,0]]};
grobjs = {pl,dt};
{t,x,y} = NumberForm[Chop[N[#,places+2]],{places+2,places},NumberPadding->{" ","0"},ExponentFunction-> (Null&) ]& /@  {t0,Sub[f1,var,t0],Sub[f2,var,t0]};
tx = Text@Row[{Style["t",Italic]," =",t,"  ",Style["x",Italic]," =",x,"  ",Style["y",Italic]," =",y}];

gr=Graphics[grobjs,Axes->True,AspectRatio->Automatic ,AxesOrigin->{Automatic,0}, ImageSize->{400,300},PlotRange->plotrange]
];

Grid[{{tx},{gr}}]
];
ParametricFunctionList = 
{{{{2t,-3t+8},{4t,3t}},{{-1,10},{-1,10}},{0,4}}->"1,2"
};
Manipulate[If[t0>Last[Last[f]],t0=Last[Last[f]]];
ParametricTrace[First[f],Prepend[Last[f],t],t0,trace,First[Rest[f]],5],
{{f,{{{2t,-3t+8},{4t,3t}},{{-1,10},{-1,10}},{0,4}},"equations"},ParametricFunctionList,BaseStyle->{10}},
{{t0,0,"t"},First[Last[f]],Last[Last[f]],Appearance->"Open"},
SaveDefinitions->True,AutorunSequencing-> {4}]

Counter Missile

A missile aimed at an air base follows the path denoted by the function {({{x=-600t+1000 }, {y=600-300t}}). The air base shoots a counter missile follows the path denoted by {({{x=150t }, {y=150t}}). Will the missiles collide? As you see below, they do...but when? and where? Prove that they collide and do not just intersect.

Sub[expr_,var_,val_]:=
expr /. var->val;
Sub[expr_,var_,vals_?ListQ]:=
Transpose[Sub[expr,var,#]& /@ vals];
ParametricTrace[{f1_,f2_},{var_,a_,b_},t0_,mode_,plotrange_,places_]:=
Module[{pl,dt,tx,gr,grobjs,t,x,y},

If[a<b,
pl= If[mode===dot || t0>a,First[ParametricPlot[{f1,f2},Evaluate[Switch[mode,dot,{var,a,b},trace,{var,a,t0},segment,{var,t0-0.1,t0+0.1}]],Axes->False,PlotStyle->Blue]],{}];
dt={PointSize[Medium],Style[Point[Sub[{f1,f2},var,t0]],RGBColor[1,.47,0]]};
grobjs = {pl,dt};
{t,x,y} = NumberForm[Chop[N[#,places+2]],{places+2,places},NumberPadding->{" ","0"},ExponentFunction-> (Null&) ]& /@  {t0,Sub[f1,var,t0],Sub[f2,var,t0]};
tx = Text@Row[{Style["t",Italic]," =",t,"  ",Style["x",Italic]," =",x,"  ",Style["y",Italic]," =",y}];

gr=Graphics[grobjs,Axes->True,AspectRatio->Automatic ,AxesOrigin->{Automatic,0}, ImageSize->{400,300},PlotRange->plotrange]
];

Grid[{{tx},{gr}}]
];
ParametricFunctionList = 
{{{{-600t+1000,600-300t},{150t,150t}},{{-1,1000},{-1,1000}},{0,5}}->"1,2"
};
Manipulate[If[t0>Last[Last[f]],t0=Last[Last[f]]];
ParametricTrace[First[f],Prepend[Last[f],t],t0,trace,First[Rest[f]],5],
{{f,{{{-600t+1000,600-300t},{150t,150t}},{{-1,1000},{-1,1000}},{0,5}},"equations"},ParametricFunctionList,BaseStyle->{10}},
{{t0,0,"t"},First[Last[f]],Last[Last[f]],Appearance->"Open"},
SaveDefinitions->True,AutorunSequencing-> {4}]

Check Your Skills

An NFL punter kicks a football downfield with initial velocity 85 feet per second at an angle of elevation of (56^o). He kicks the ball from a height of 3 feet.

Journal: Write an equation for and draw a complete graph of the problem.

Journal: How far downfield will the football first hit the field?

Journal: Determine the maximum height of the ball above the field.

Journal: What is the “hang time”? (That is, the total time the football is in the air.)

_______________________________________________________________________________________________________________________________________

 Assume two particles are travelling on the same 2 dimensional plane. The paths of the two particles are shown below.

Particle 1 is {({{x=t}, {y=(t^2)}})       Particle 2 is {({{x=(1/2)t+1}, {y=2t}})

Journal: Do the particles collide? If so, at what time?