Rate of Change of Shadow

If I understand the problem correctly, this is simple geometry of two triangles. Even my dog can do this one because she likes to play with shadows. The first triangle is the triangle made from the three points, which are the top of the pole, the bottom of the pole, and the light point source.

The second triangle is the top of the pole, the bottom of the pole, and the end of the shadow. The line with the end points that contain the point light source and the shadow's end point also contains the point that represents the flag pole.

You will need to know the angle of incidence of this line relative to the flag pole and the length of the pole, or you need to know the lengths and angles of the two triangle's legs so that you can reconstruct and solve the problem.

To make it easier you could assume (if this is the case) that the angle the pole makes to the line that represents its shadow can be 90 degrees.

Still, you can now solve this using basic trigonometry, which could be the lengths of the triangle sides and/or the sin/cosines of pertinent angles. Ultimately you can write a trigonomic function that describes the shadow's length given the relative angle of the light point source to the pole and the height of the pole. That function would express the shadow length as a function of the pole height and the tangent of the light source to the pole. The function will have an output that has limits that run from +/- infinity for relative angles of 0 to 180 degrees. Angles beyond this are meaningless. You see? This is getting easier and easier.

Regardless of the length of the pole, the shadow length can be expressed as a ratio of the length of the pole. When the light is at 0 degrees the length of the shadow is infinite. When at 45 degrees the ratio is 1. At 90 degrees it is 0. Hmmm. Sounds like the inverse of the tangent of the angle that the light source makes with the pole's tip. Pretty simple. Extending the angle beyond 90 degrees (high noon) produces a negative shadow length, which is simply a shadow in the opposite direction.

If the pole is not at a right angle to the shadow that it casts, then things get more complex. I'll leave that up to you to work out. ;-)

You just need to express the length of shadow as function of height of light source (simple geometry of similar triangles) and differentiate.

L = length of shadow

D = distance on ground from starting point of light source to base of pole

H = height of pole

v = velocity of light source

Minus sign because shadow gets shorter as h increases. Check - when h = H, L and

dL/dt = ∞, when h = ∞, L and dL/dt = 0.