Problem 11 - Physics

Question

A spotlight on a boat is \(2.5 \mathrm{~m}\) above the water, and the light strikes the water at a point that is \(8.0 \mathrm{~m}\) horizontally displaced from the spotlight (see the drawing). The depth of the water is \(4.0 \mathrm{~m}\). Determine the distance \(d\), which locates the point where the light strikes the bottom.

Step-by-Step Solution

Answer

The light strikes the bottom at 10.0 m horizontally from the spotlight.

Step 1: Understand the Geometry

Visualize the scenario as a right triangle, where the vertical distance from the light on the boat to the water's surface is one side (2.5 m), and the horizontal distance to the point where light hits the water is another side (8.0 m).

Step 2: Identify the Path of Light

The light travels the hypothenuse of the triangle formed between the vertical (2.5 m) and horizontal (8.0 m) distances.

Step 3: Calculate the Angle of Incidence

Use trigonometry to find the angle of incidence. The angle can be calculated as \( \theta = \tan^{-1}\left(\frac{2.5}{8.0}\right) \).

Step 4: Calculate the Position of Light on the Bottom

For the submerged region, apply Snell's Law to find the refraction angle. Then, calculate the horizontal distance under water. The total horizontal displacement \(d = 8.0 + (4.0 \tan \theta_r)\), where \(\theta_r\) is the angle of refraction.

Step 5: Determine Final Distance

Add the horizontal distances before and after the water surface to find the total distance light travels from the spotlight to the point on the bottom.

Key Concepts

Right Triangle

In many geometry problems, the right triangle plays a crucial role due to its well-defined properties. A right triangle has one angle that is exactly 90 degrees. This structural characteristic provides a foundation for various calculations, such as determining unknown lengths or angles.
In our scenario, we imagine a right triangle where:

• Vertical side: The distance from the spotlight to the water surface, which is 2.5 meters.
• Horizontal side: The distance from the point directly below the light to the spot where the light strikes the water, which is 8.0 meters.
• Hypotenuse: The path that the light travels, from the boat to the point at the water surface.

This geometric setup allows us to apply trigonometric principles to find information like angles and other lengths within the triangle.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, especially right triangles. It is incredibly useful in various fields, including navigation and physics.
In the context of our problem, trigonometry is employed to determine the angle of incidence, a key element for subsequent calculations. To find the angle of incidence, we make use of the tangent function:

• The tangent function relates an angle in a right triangle to the ratio of the length of the opposite side to the length of the adjacent side.
• We use \( \theta = \tan^{-1}\left(\frac{2.5}{8.0}\right) \) to calculate the angle of incidence.

Trigonometric relations offer a pathway from knowing side lengths to determining angles, making them fundamental in solving geometry-related physics problems.

Snell's Law

Snell's Law explains how light bends when it passes from one medium into another, such as from air to water. The law is expressed as:\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]where \( n_1 \) and \( n_2 \) are the refractive indices of the original and new mediums, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction, respectively.
In our problem, Snell's Law is applied after determining the initial angle as the light transitions from air into water. With water having a different refractive index from air, the light bends, resulting in a new angle, \( \theta_r \).
Knowing the angle of refraction is crucial for calculating how far horizontally the light travels beneath the water, which leads directly to determining the final strike point on the bottom surface.

Refraction

Refraction is the bending of light as it passes from one medium to another with a different density, which changes its speed. In our exercise, when the spotlight beam enters the water, it refracts or bends due to this effect.
The extent to which light rays bend is governed by the indices of refraction for the involved media. This means when light passes from air into water, its speed decreases, causing the beam to shift direction.
This concept is visually evident as the angle of refraction, \( \theta_r \), differs from the angle of incidence. Understanding refraction allows us to trace the exact path of the light underwater, pivotal in determining where the beam ultimately impacts the bottom of the sea.

Angle of Incidence

The angle of incidence is the angle between the incoming light and a line perpendicular to the surface it strikes. In physics and optics, this measurement is essential for predicting how light will behave at the boundary between two different materials.
In our geometry problem, the angle of incidence is found using trigonometric calculations from the right triangle formed by the boat, the water's surface, and light path.
By identifying this angle, we can apply Snell's Law to determine the angle of refraction, which tells us how the light's path changes upon entering a new medium. This angle is crucial, as it informs the overall trajectory from the surface down to the seabed.