Motion of a Shadow

To be able to calculate an object’s velocity using geometry and calculus. (Figure 1) A light source, S, is located at a distance l from a wall. An opaque object of height h moves toward the wall with a constant velocity v. At time t=0 , the object is located at the source S.

Expression for the magnitude of velocity of shadow Find an expression for vs, the magnitude of the velocity vs of the top of the object's shadow, at time t. Express the speed of the top of the object's shadow in terms of t, v, l, and h.

This problem can be done with ratios

There are two right triangles. One is formed by V*t and h. V*t is the distance the wall traveled from s h is the height of the wall. This form a ratio:

ratio = h/V*t

Now there is the second triangle, which is formed by l and V_S*t which is the distance the shadow traveled. This has the same ratio as the first triangle.

ratio₂ = V_S*t/l

Since the ratios are equal we place them in an equation and solve for V_S,

ratio = ratio₂

h/V*t = V_S*t/l

Solve for V_S

V_S = h*l/V*t²