Shadow Length’s Rate of Increase: A Mathematical Puzzle

Engaging Minds with Simple yet Challenging Problems

When a person of height 2 meters walks away from a 6-meter lamp post at a uniform speed of 5 km/h, the question of how the length of their shadow changes with time presents a fascinating puzzle. This problem, while seemingly simple, requires a blend of geometric thinking and calculus. Such questions are a stark contrast to common beliefs about the flat Earth and anti-relativity theories that often dominate the internet. This article delves into the math behind this engaging problem.

A Geometric Representation

The situation can be visualized using a geometric diagram. Consider the lamp post as point A and the man as point C. At any point in time t, the man stands at point D, and the tip of his shadow is at point E. The distance between the lamp post and the man is denoted by x, and the length of the shadow is denoted by l. We can use similar triangles to solve the problem.

Similar Triangles

From the similar triangles ΔEDC and ΔEBA, we have the proportion:

( frac{ED}{CD} frac{EB}{AB} )

Substituting the given values, we get:

( frac{l}{2} frac{x - l}{6} )

From this, we can derive:

( frac{x - l}{l} frac{6}{2} 3 )

( 1 - frac{l}{x} 3 )

( frac{l}{x} 2 )

( l frac{x}{2} )

Rate of Change Analysis

By differentiating both sides of the equation with respect to time t, we get:

( frac{dl}{dt} frac{d}{dt} left( frac{x}{2} right) )

( frac{dl}{dt} frac{1}{2} times frac{dx}{dt} )

From the problem, we know that dx/dt 1 m/s because the person is walking at a constant speed. Substituting this, we obtain:

( frac{dl}{dt} frac{1}{2} times 1 )

( frac{dl}{dt} 0.5 text{ m/s} )

Thus, the rate at which the length of the shadow increases is 0.5 m/s.

Velocity of the Shadow Tip

Now, let’s consider the velocity of the tip of the shadow E relative to the lamppost. Assuming an inertial frame of reference fixed with the lamppost, we can express the relative velocity as:

( vec{v}_{BE} vec{v}_{BD} - vec{v}_{DE} )

With ( vec{v}_{BD} 1 text{ m/s} ) (since the man is moving away from the lamp post at a constant speed), and ( vec{v}_{DE} 0.5 text{ m/s} ) (the rate of increase of the shadow length), we get:

( vec{v}_{BE} 1 - 0.5 1.5 text{ m/s} )

This means that the tip of the shadow is moving at a speed of 1.5 m/s relative to the lamppost.

Concluding Remarks

This problem highlights the importance of basic mathematical principles in solving real-world problems. It also emphasizes the importance of adhering to scientific principles and understanding over misconceptions. Solving such problems can be a great way to stimulate interest in mathematics and physics among students and enthusiasts.