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### one in vacuum and one in air

H.Tahsiri

```Off[General::spell];
Off[General::spell1];```

`Clear["Global`*"];`

`Clear[vequ,initvel,vsol,yequ,inity,ysol,v,y,v0,y0,time,k,yna,vna,t,given,p,g]`

```vequ=v'[t]==-g+k v[t]^2

```

`vsol=DSolve[{vequ,v[0]==0},v[t],t];`

```
```

`ysol=DSolve[{yequ,y[0]==0},y[t],t];`

```
```

```tsol=Solve[y==y[t],t]

```

```k=3 c roha/(8 R rohw)

```

```given={c->.6,roha->1.2,rohw->10^3,R->1.5 10^(-3),g->9.8,y->-1200}

{c->0.6,roha->1.2,rohw->1000,R->0.0015,g->9.8,y->-1200}```

```tdown=t/.tsol[[2]]/.given

163.153```

```impactspeed=y'[t]/.given/.t->tdown

-7.37865```

`velheightplot=ParametricPlot[Evaluate[{y'[t]  ,y[t]}/.given,{t,0,7},PlotRange->All]];`

`(* RAIN DROPS IN THE ABCENCE OF AIR ; ARE WE LYING TO OUR STUDENT? YES OR NO  *)`

`Clear[yna,vna,t]`

```yna[t_]=-(1/2) g t^2

```

```tna=Sqrt[-2y/g]/.given

15.6492```

```vna[t_]=D[yna[t],t]

-g t```

`velheightplotnoair=ParametricPlot[Evaluate[{vna[t]  ,yna[t]}/.given,{t,0,7},PlotRange->All]];`

`togrther=Show[{velheightplot,velheightplotnoair},PlotRange->{{0,-15},{0,-20}}];`

`Clear[p,g,t,air,noair]`

```p[t_]:=p[t]=
Show[Graphics[{
PointSize[.04],Point[{1,y[t]/.given}],
PointSize[.04],Point[{3,yna[t]/.given}]}],
PlotRange->{{0,5},{-40,2}},Axes->Automatic,Ticks->{None,Automatic}];```

`g=Do[p[t],{t,0,7,.2}]`