Problem 4 - Physics

Question

The drawing shows a laser beam shining on a plane mirror that is perpendicular to the floor. The beam's angle of incidence is \(33.0^{\circ} .\) The beam emerges from the laser at a point that is \(1.10 \mathrm{~m}\) from the mirror and \(1.80 \mathrm{~m}\) above the floor. After reflection, how far from the base of the mirror does the beam strike the floor?

Step-by-Step Solution

Answer

The beam strikes the floor 3.87 meters from the base of the mirror.

Step 1: Identify The Given Information

Given in the problem are the angle of incidence \( \theta_i = 33.0^{\circ} \), the initial horizontal distance from laser to mirror \( d = 1.10 \text{ m} \), and the vertical height of the laser from the floor \( h = 1.80 \text{ m} \). The angle of reflection is equal to the angle of incidence, \( \theta_r = 33.0^{\circ} \), due to the law of reflection.

Step 2: Determine the Path of the Reflected Beam

The beam will strike the floor at a distance \( x \) from the base of the mirror. This distance can be found using the tangent of the angle of reflection \( \theta_r \), where \( \tan(\theta_r) = \frac{\text{opposite}}{\text{adjacent}} = \frac{1.80 \; \text{m}}{x} \).

Step 3: Solve for Distance x

Rearrange the equation from Step 2 to solve for \( x \):\[ x = \frac{1.80 \; \text{m}}{\tan(33.0^{\circ})} \] Calculate \( x \):\( \tan(33.0^{\circ}) \approx 0.6494 \)\[ x = \frac{1.80}{0.6494} \approx 2.77 \; \text{m} \]

Step 4: Calculate Total Distance from Laser to Floor Impact

The total horizontal distance from the initial point of the laser to where it hits the floor can be obtained by adding the initial distance from the laser to the mirror and the calculated distance \( x \):\[ \text{Total Distance} = 1.10 + 2.77 = 3.87 \; \text{m} \]

Key Concepts

Understanding Angle of Incidence

The angle of incidence is a critical concept in understanding how light behaves when it encounters a surface. It is defined as the angle between the incoming ray of light and the normal (an imaginary line perpendicular) to the surface at the point of contact. In the context of the problem, the laser beam makes a specific angle, known as the angle of incidence, with the perpendicular mirror. Here, this angle is given as \(33.0^{\circ}\).
You can imagine the normal as if you are standing right where the light hits the mirror and holding a stick straight out! Knowing this allows you to understand more about how the light will reflect.
This angle is fundamental in determining how the light will reflect off the surface, following the rules of the "law of reflection." Understanding this angle helps in predicting the path the light will take, which is crucial for solving physical problems involving reflections.

The Law of Reflection Simplified

The law of reflection is a simple yet powerful rule that dictates how light behaves upon encountering reflective surfaces, like mirrors. Simply put, the law states that the \(\text{angle of incidence}\) \((\theta_i)\) is equal to the \(\text{angle of reflection}\) \((\theta_r)\). Therefore, if a beam strikes a mirror at \(33.0^{\circ}\) to the normal, it reflects off the mirror at the same angle.
This rule is consistent for all types of reflections and helps predict exactly where the light will go.

• It doesn't matter what kind of mirror or the color of the light, as long as the surface is smooth and even, like the plane mirror in the problem.
• This principle is helpful in solving problems where light paths need to be traced, such as the scenario presented, where we need to understand how the laser sees its path change from mirror to floor.
  By applying this law, it is straightforward to map out the light's journey beyond just theoretical understanding.

Applying the Tangent Function

The tangent function is a trigonometry function commonly used in problems involving right triangles. It is defined as the ratio of the opposite side to the adjacent side for a given angle in a right triangle.
In the context of this light reflection problem, we use tangent to determine the additional distance the laser beam travels after reflecting off the mirror.
For the angle of reflection \(33.0^{\circ}\):
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{height from floor}}{\text{distance from mirror base, } x} \]
Simplifying and solving this equation allowed us to find the additional distance \(x\) the light travels horizontally on the floor, crucial to determining the precise landing spot of the reflected beam.
This demonstrates how trigonometry, specifically the tangent function, can visually and mathematically describe physical events.

Physics Problem Solving Made Easy

Physics problems often seem intimidating at first, but breaking them down is the key to revealing simple solutions. The laser beam reflection problem is a great example of how to approach these types of questions methodically.
The key steps in physics problem-solving typically include:

• **Identify what is given and what you need to find**: Separate known values (like angles and distances) from unknowns (like where the beam hits the floor).
• **Apply relevant theories and formulas**: Use established laws like the law of reflection and mathematical functions like tangent to relate these concepts together.
• **Carry out the calculations**: Plug in the numbers and solve. Double-check your calculations for accuracy.
• **Interpret and validate results**: Ensure that your solutions are realistic within the physical context.

Approaching problems this way makes them more manageable and teaches valuable skills for solving many real-world physical phenomena.