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H.Tahsiri

UP AND DOWN MOTION OF A BALL IN VACCUM AND AIR

Please follow the program below to see how the above animation is done

UP AND DOWN MOTION OF A BALL IN VACCUM

Clear[y,t,given,v0,g,y0]

given1={v0->50,g->9.8,y0->0}

     {v0 -> 50, g -> 9.8, y0 -> 0}

y=y0+v0 t-.5 g t^2

             2
     -0.5 g t  + t v0 + y0

ynoair = y/.given1

                 2
     50 t - 4.9 t

totaltnoair=Solve[ynoair==0,t]

     {{t -> 0.}, {t -> 10.2041}}

yupdownoairplot=Plot[ynoair,{t,0,10.2041}];

UPWARD MOTION OF A BALL IN AIR

Clear[y,v]

given2={b->.1,g->9.8,v0->50}

     {b -> 0.1, g -> 9.8, v0 -> 50}

aupair=-g -b v[t]

     -g - b v[t]

vupair=DSolve[{v'[t]==aupair,v[0]==v0},v[t],t]

                    b t
               g - E    g + b v0
     {{v[t] -> -----------------}}
                       b t
                    b E

vupairsymbl=vupair[[1,1,2]]

          b t
     g - E    g + b v0
     -----------------
             b t
          b E

vupairnum=vupairsymbl/.given2

                      0.1 t
     10. (14.8 - 9.8 E     )
     -----------------------
              0.1 t
             E

tupair=Solve[vupairnum==0,t]

     {{t -> 4.12245}}

tupairvalue=tupair[[1,1,2]]

     4.12245

yupair=DSolve[{y'[t]==vupairsymbl,y[0]==0},y[t],t]

                     b t        b t                 b t
               -g + E    g - b E    g t - b v0 + b E    v0
     {{y[t] -> -------------------------------------------}}
                                  2  b t
                                 b  E

yupairsymbl=yupair[[1,1,2]]

           b t        b t                 b t
     -g + E    g - b E    g t - b v0 + b E    v0
     -------------------------------------------
                        2  b t
                       b  E

tocheck=Limit[yupairsymbl,b->0]

          2
     -(g t )
     ------- + t v0
        2

yupairnum=yupairsymbl/.given2

                         0.1 t         0.1 t
     100. (-14.8 + 14.8 E      - 0.98 E      t)
     ------------------------------------------
                        0.1 t
                       E

yupvalue=yupairnum/.t->tupairvalue

     96.0001

yupairtable=Table[{t,yupairnum},{t,0,tupairvalue,.09}];

yupairplot=ListPlot[yupairtable,PlotJoined->True];

DOWNWARD MOTION OF A BALL IN AIR

Clear[v,t,y]

given3={b->.1,g->9.8,v0->50}

     {b -> 0.1, g -> 9.8, v0 -> 50}

adownair=g-b v[t]

     g - b v[t]

vdownair=DSolve[{v'[t]==adownair,v[0]==0},v[t],t]

               g     g
     {{v[t] -> - - ------}}
               b      b t
                   b E

vdownairsymbl=vdownair[[1,1,2]]

     g     g
     - - ------
     b      b t
         b E

vdownairnum=vdownairsymbl/.given3

            98.
     98. - ------
            0.1 t
           E

ydownair=DSolve[{y'[t]==vdownairsymbl,y[0]==0},y[t],t]

                 g        g      g t
     {{y[t] -> -(--) + ------- + ---}}
                  2     2  b t    b
                 b     b  E

ydownairsymbl=ydownair[[1,1,2]]

       g        g      g t
     -(--) + ------- + ---
        2     2  b t    b
       b     b  E

ydownairnum=ydownairsymbl/.given3

              980.
     -980. + ------ + 98. t
              0.1 t
             E

tdownair=FindRoot[ydownairnum==96.0001,{t,4}]

     {t -> 4.77834}

tdownairvalue=tdownair[[1,2]]

     4.77834

tupairvalue

     4.12245

ydownairtable=Table[{t+tupairvalue,96.0001-ydownairnum},
{t,0,tdownairvalue,.09}];

ydownairplot=ListPlot[ydownairtable,PlotJoined->True,
AxesOrigin->{tupairvalue,0}];

UP AND DOWN MOTION TOGETHER

yupdownairtable=Join[yupairtable,ydownairtable];

yupdownairplot=ListPlot[yupdownairtable,PlotJoined->True];

together=Show[{yupdownairplot,yupdownoairplot}];

SIMULATION OF AN UP AND DOWN MOTION OF A BALL IN VACCUM AND AIR

ynoair
                 2
     50 t - 4.9 t

g=Table[{0,ynoair},{t,0,10.2,.1}];

Length[g]

     103

gg=Table[{t,ynoair},{t,0,10.2,.1}];

gair=Table[{0,yupdownairtable[[i,2]]},{i,1,100}];

ggair=Table[yupdownairtable[[i]],{i,1,100}];

Length[gair]

     100

Length[ggair]

     100

Do[
Show[(*(Graphics[Dashing[{.008}],Thickness[.003],Line[{g[[n]],gg[[n]]}]])*)

ListPlot[gg,PlotJoined->True,PlotRange->{{0,10.20},{0,150}},
PlotStyle->{Thickness[.01],RGBColor[0,1,0]},DisplayFunction->Identity],

ListPlot[{gg[[3n]]},PlotRange->{{0,10.20},{0,150}},
PlotStyle->{RGBColor[1,0,0],PointSize[.02]},
DisplayFunction->Identity],

ListPlot[{g[[3n]]},PlotRange->{{0,10.20},{0,150}},
PlotStyle->{RGBColor[1,0,1],PointSize[.05]},
DisplayFunction->Identity],


ListPlot[ggair,PlotJoined->True,PlotRange->{{0,10.20},{0,150}},
PlotStyle->{Thickness[.01],RGBColor[0,1,0]},DisplayFunction->Identity],


ListPlot[{ggair[[3n]]},PlotRange->{{0,10.20},{0,150}},
PlotStyle->{RGBColor[1,0,0],PointSize[.02]},
DisplayFunction->Identity],


ListPlot[{gair[[3n]]},PlotRange->{{0,10.20},{0,150}},
PlotStyle->{RGBColor[1,0,1],PointSize[.05]},
DisplayFunction->Identity],

DisplayFunction->$DisplayFunction],   (* end Show *)

{n,1,33}]    (* end Do  *)


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