Kinematics
Advanced
linear motion
Source: High School Physics Olympiad
Problem Sets:
kinetics - olympiad
Problem
A street lamp is at height $H$ above the ground. A pedestrian of height $h$ walks away from the lamp at uniform speed $v$.
At what speed does the shadow of the person's head move along the ground?
$v_{shadow} = \dfrac{Hv}{H - h}$
Let $S$ be the person's distance from the base of the lamp and $x$ the distance of the head's shadow from the base. The light ray from the lamp top through the person's head reaches the ground at the shadow, so by similar triangles:
$$\frac{h}{H} = \frac{x - S}{x}$$Solving for $x$:
$$x = \frac{H}{H - h}S$$Differentiating with respect to time (the factor $\frac{H}{H-h}$ is constant):
$$v_{shadow} = \frac{H}{H - h}\,\frac{dS}{dt} = \frac{Hv}{H - h}$$Since $H > h$, the shadow moves faster than the person.