Speed of Tip of Shadow Relative to Person

To find the speed of the tip of the person's shadow with respect to the person, let:

• $H = 4\text{ m}$ be the height of the lamp post.
• $h = 1.6\text{ m}$ be the height of the person.
• $x$ be the distance of the person from the base of the lamp post.
• $s$ be the length of the person's shadow, which represents the position of the tip of the shadow with respect to the person.

From the geometry of similar triangles formed by the light source, the person, and the ground: $\frac{H}{x + s} = \frac{h}{s}$

Cross-multiplying and rearranging the terms gives: $H s = h(x + s)$ $(H - h) s = h x$ $s = \left(\frac{h}{H - h}\right) x$

Substitute the given values for $H$ and $h$: $s = \left(\frac{1.6}{4 - 1.6}\right) x = \left(\frac{1.6}{2.4}\right) x = \frac{2}{3} x$

Differentiating both sides with respect to time $t$: $\frac{ds}{dt} = \frac{2}{3} \frac{dx}{dt}$

Here, $\frac{dx}{dt} = 60\text{ cm s}^{-1}$ is the speed of the person walking away from the lamp post, and $\frac{ds}{dt}$ represents the speed of the tip of the shadow with respect to the person.

$\frac{ds}{dt} = \frac{2}{3} \times 60\text{ cm s}^{-1} = 40\text{ cm s}^{-1}$

Therefore, the speed of the tip of the person's shadow on the ground with respect to the person is 40.