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202202080958 Dynamics Figures (Elementary) Q5

2. A man walking with a speed v constant in magnitude and direction passes under a lantern hanging at a height H above the ground. Find the velocity which the edge of the shadow of the man’s head moves over the ground with if his height is h.

Extracted from B. Bukhovtsev et al. (1978). Problems in Elementary Physics.

Solution.

Let x=0 be the position of the lantern; let the man walk in the positive x-direction; and let the position of the man be x_m(t) and that of the shadow of his head x_s(t). So the length s of his shadow is |x_s-x_m|.

By comparing similar triangles, we have

\displaystyle{\frac{H}{x_s} = \frac{h}{s}}.

Thus,

\begin{aligned} \frac{H}{x_s} & = \frac{h}{x_s-x_m} \\ x_s & = \bigg(\frac{H}{H-h}\bigg) x_m \\ \dot{x}_s & = \bigg(\frac{H}{H-h}\bigg) \dot{x}_m \\ \dot{x}_s & = \bigg(\frac{H}{H-h}\bigg) v \\ \end{aligned}

\therefore The edge of the shadow of the man’s head moves with a velocity (\frac{H}{H-h}) v\,\hat{\mathbf{i}} over the ground.