Exercise 6: Galileo's experiment

According to an apocryphal tale, Galileo dropped a 100 pound cannonball and a 1 pound ball from the Leaning Tower of Pisa in order to disprove the Aristotelian view that objects fall with velocities proportional to their weights. If the objects hit the ground at the same time, then the Aristotelian school was wrong. Is the no-drag model consistent with Galileo's supposed observation that the balls hit the ground at the same time? Is the drag model? In actuality, Galileo tried a similar experiment in his youth. He dropped a lump of lead and a lump of wood from a high tower. Suppose the tower was 50 meters high, the lead was a sphere of diameter 2 cm, the wood was a sphere of radius 10 cm, and the air temperature was 20oC. The density of lead is 11.3 gm/cm3 and the density of wood, say pine, is 0.5 gm/cm3. Use the no-drag model to predict the time elapsed before each object hits the ground. Repeat for the drag model.

You can create an animation of two falling spheres, from t = 0 to t = tmax by executing the following command. You will need to replace the definitions of y1 and y2 given below with your own definitions. You will also need to replace tmax.

In[52]:=

  Clear[t,d,rho]
  
  g = 980;
  k2 = Pi d^2 rho/16;
  tmax = 3.5;
  mwood = 4/3 Pi 5^3 0.5;
  k = k2 /. {d -> 10, rho -> 1.21 10^-3};
  y0 = 5000; v0 = 0;
  y1 = -1/2 g t^2 + v0 t + y0;
  y2 = -m/k Log[Abs[Cosh[-Sqrt[k g/m] t]]] + y0 /. m -> mwood;

Here is the animation.

In[53]:=

  <<Graphics`Master`
  Clear[time]
  rad = 0.1;
  Do[
     yposition1 = Max[Evaluate[y1 /. t -> time],0];
     yposition2 = Max[Evaluate[y2 /. t -> time],0];
     corners = Graphics[{RGBColor[1,1,1],
            Point[{0,-rad/2}], Point[{1,-rad/2}],
            Point[{0,y0*(1+rad/2)}],Point[{1,y0*(1+rad/2)}]}];
     spheres = Graphics[{RGBColor[0,0,1],PointSize[0.1],
            Point[ {0.3,yposition1}],
            Point[ {0.7,yposition2}]},
            PlotRange -> {{0,1},{-rad/2,y0+rad/2}}];
     text1 = Graphics[Text["y1", Scaled[{0.15,0.9}]]];
     text2 = Graphics[Text["y2", Scaled[{0.85,0.9}]]];
     Show[text1, text2, corners, spheres, Frame -> True];,
  {time,0,tmax,tmax/10}]


(This is the first frame of a 48996 byte mpg movie. Click on the graph to download the movie)

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