Me and My Shadow - the former on the ground the latter on a dome
BRIEF ABSTRACT
We seek to characterize the motion of my shadow on a dome from a light
source on the ground behind me as I walk toward the dome.
GENERAL INFORMATION
FileName:   SHADOW

Full title:  Me and My Shadow - the former on the ground the latter on a dome.

Developer:  
Brian J. Winkel, Department of Mathematics, 
Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. 

Contact:  
Brian J. Winkel, Department of Mathematics, 
Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. 
Phone:  812-877-8412.  Email:  winkel@rose-hulman.edu.  
FAX:  812-877-3198.

Support:  
The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849:  Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. 
STATEMENT OF PROBLEM 
I walk the bumpy path to the small athletic dome where I work out -- a spherical dome of radius 10 m centered at the origin (0, 0).  I stand 2.5 m tall (I play for the Boston Celtics in high heel sneakers!!!) and I  walk along the wavy path given by the equation y = sin(x + 30) m.  I start  at point (-30, 0) m.  I move at a horizontal speed of 1 m/sec on a straight line path along the x-axis toward the dome as I go up and down the wavy path.  A lawn walk light sitting at position (-50, 0) m casts a shadow of the top of my head on the wall of the bubble.
What is the maximum speed the top of the shadow attains as I walk up to the dome as described?
KEYWORDS  
Parametric equation description of motion, linear function,  angular velocity, radial velocity, maximization.

TEACHER NOTES
ISSUES RELATED TO THE PROBLEM
Prerequisites
Parametric equation description of motion, linear function,  angular velocity, radial velocity, maximization.
Time allotment - time management
This problem can really eat up time because of the lttle pieces involve, but the essence is setting up the functionx properly, working with the line of light, and determining where this line of light falls on the dome - ALL as functions of time.
But if the instructor gives appropriate guidance in class it can be essentially compleely introduced and set up in one 45 minute class with time after that for outside class write up or a portion of some near future class time.
For example, the instructor could drawe a sketch of the location and the position of the shadow at several points in time, might attempt to identify variables, relate variables, and do some of the geometry and trigonometry useful to get a mathematical problem formulation.
Expectations 
We expect the student to put togehter basic pieces to solve this complex problem
Future payoffs
Extensions
This problem IS an extension of the classical problem in which we are concerned about the shadow on a vertical wall while one walks to or from the wall.
Possible further extensions would be different surfaces on which the shadow is projected, different rules of motion for the walker, OR a moving light source.
Starting closer to the light source so that the tip of the shadow goes off the dome for some time t might be interesting.
References and Sources
This is a nice quote I have used with this material.

One can live in the shadow of an idea without grasping it.
The Heat of the Day (1949) ELIZABETH BOWEN 1899Ð1973
POSSIBLE SOLUTION(S)
We plot some notions which will help us graphically in this problem.

c = ParametricPlot[{10 Cos[t],10 Sin[t]},{t,0, 2 Pi}, 
	PlotRange->{0,12}, AspectRatio->Automatic]
s = Plot[  Sin[x+30] + 2.5,{x,-30,-9.7},PlotRange->{-12,12},
	AspectRatio->Automatic]
Show[s,c,AspectRatio->Automatic,
	PlotRange->{{-50,12},{-12,12}}]
We denote the position (xp(t), yp(t)) of the top of my head as a function of time, t. 
xp[t_] = -30 + t
-30 + t
yp[t_] = 2.5  + Sin[t]
2.5 + Sin[t]
We determine the time the top of my head arrives at the dome.
sol = FindRoot[xp[t]^2 + yp[t]^2==100,{t,6 Pi}]
{t -> 20.6248}
tf = t/.sol[[1]]
20.6248
We now attempt to find the location (xt[t], yt[t]) of the shadow on the dome at time t.
First, the slope of the line of light from the light (-50, 0 ) to the top of my head (xp[t], yp[t]).
f[x_] = (yp[t])/(xp[t]+50) (x + 50)
(50 + x) (2.5 + Sin[t])
-----------------------
        20 + t
eq = x^2 + f[x]^2 == 100
             2               2
 2   (50 + x)  (2.5 + Sin[t])
x  + ------------------------- == 100
                     2
             (20 + t)
We determine where the line of light intersects the dome.
sol = Solve[eq,x];
xt[t_] = x/.sol[[1]];
yt[t_] = Sqrt[100 - xt[t]^2];
Here we animate the motion of the top of my head and the shadow on the dome.
Do[
	Show[s,c,
		Graphics[{PointSize[.02],Point[{xt[t],yt[t]}],
			Point[{xp[t],yp[t]}],Point[{-50,0}],
		Line[{{xt[t],yt[t]},{-50,0}}]},
			AspectRatio->Automatic,
			PlotRange->{{-50,12},{-12,12}}
				]],{t,0, tf, .25}]
Now to calculute the speed of the shadow on the dome. 
First we calculate the angle subtended by the arc along the dome from the point ((x(0), y(0)) to (x(t), y(t))
T[t_] = ArcTan[yt[t]/xt[t]] - ArcTan[yt[0]/xt[0]];
Plot[T[t],{t,0,tf}]
Next we need to calculate the speed of the tip of the shadow along the circle and this can be obtained by (recall the radius is 10 m) multiplint the angular velocity in radians by the radius. 
speed[t_] = 10 T'[t];
Plot[speed[t],{t,0,tf}]
And eventually the maximum speed and the time at which this occurs.  We find the maximum speed of the tip of the shadow is 2.444 m/sec and this occurs at time t = 2.76873 sec after I start my walk toward the dome.
mspeed[t_] = speed'[t];
solt = FindRoot[mspeed[t]==0,{t,3}]
{t -> 2.76873}
speed[t]/.solt[[1]]
2.44471
ISSUES IN SOLUTION
This problem could be regarded as a complicated, long problem, but if taken one step at a time the compicated appearance settles down to a reasonable extension of the traditional problem in which the shadow falls on a straight wall.