A person of height 1.6 m is walking away from a lamp post of height 4 m along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is 60 cm/s, the speed of the tip of the person’s shadow on the ground with respect to the person is ________cm/s.

The correct answer is 40.

Step 1: Understand the Problem and Setup the Geometry

Let the height of the lamp post be $H=4\text{m}$ and the height of the person be $h=1.6\text{m}$.

Let the lamp post be situated at the origin on a straight path along the ground.

Let $x$ be the distance of the person from the lamp post at any time $t$.

Let $s$ be the length of the person's shadow cast on the ground.

The total distance of the tip of the shadow from the base of the lamp post is given by $y=x+s$.

Step 2: Use Similar Triangles

Since both the lamp post and the person stand perpendicular to the flat ground, they form two similar right-angled triangles.

By the properties of similar triangles, the ratio of their heights is equal to the ratio of the distances from the tip of the shadow:

$\frac{H}{h}=\frac{x+s}{s}$

Substitute the given values for $H$ and $h$:

$\frac{4}{1.6}=\frac{x+s}{s}$

Simplifying the fraction $\frac{4}{1.6}=2.5=\frac{5}{2}$:

$\frac{5}{2}=1+\frac{x}{s}$

$\frac{x}{s}=\frac{5}{2}-1=\frac{3}{2}$

Cross-multiplying gives the relationship between the length of the shadow $s$ and the position of the person $x$:

$s=\frac{2}{3}x$

Step 3: Calculate the Required Speed

We are given that the speed of the person is:

$v_{\text{person}}=\frac{dx}{dt}=60\text{cm/s}$

The speed of the tip of the shadow with respect to the person is simply the rate of increase of the length of the shadow, which is $\frac{ds}{dt}$.

Differentiating $s=\frac{2}{3}x$ with respect to time $t$:

$\frac{ds}{dt}=\frac{2}{3}\frac{dx}{dt}$

Substitute $\frac{dx}{dt}=60\text{cm/s}$:

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

Conclusion:

The speed of the tip of the person's shadow with respect to the person is 40 cm/s.