Problem 41 - Physics

Question

A lamp is twice as far in front of a plane mirror as a person is. Light from the lamp reaches the person via two paths, reflected and direct. It strikes the mirror at a \(30.0^{\circ}\) angle of incidence and reflects from it before reaching the person. The total time for the light to travel this path includes the time to travel to the mirror and the time to travel from the mirror to the person. The light also travels directly to the person without reflecting. Find the ratio of the total travel time along the reflected path to the travel time along the direct path.

Step-by-Step Solution

Answer

The ratio of the travel times is 3:1.

Step 1: Understand Path Distances

Let the distance from the lamp to the mirror be \( d \) and from the person to the mirror be \( d/2 \). Thus, the direct path distance is \( d/2 \). We know the reflected path distance involves going to the mirror (\( d \)) and then to the person (\( d/2 \)).

Step 2: Calculate Reflected Path Distance

The total reflected path distance is the sum of the path to the mirror and the path to the person: \( d + \frac{d}{2} = \frac{3d}{2} \).

Step 3: Calculate Direct Path Distance

The direct path distance is simply \( d/2 \).

Step 4: Calculate Travel Times

Assuming the speed of light is \( c \), the travel time along the reflected path is \( \frac{3d}{2c} \) and along the direct path is \( \frac{d}{2c} \).

Step 5: Find Ratio of Reflected to Direct Travel Times

The ratio of the travel times is given by the formula\[ \frac{\text{Reflected Time}}{\text{Direct Time}} = \frac{\frac{3d}{2c}}{\frac{d}{2c}} = \frac{3}{1} = 3. \]

Key Concepts

Angle of Incidence

The angle of incidence is a fundamental concept in geometric optics that helps us understand how light behaves when it strikes a surface. When light hits a surface, it does so at a certain angle in relation to the normal - an imaginary line perpendicular to the surface. In this scenario, the light from the lamp hits the mirror at an angle of incidence of \(30^{\circ}\). This angle determines how the light will reflect off the mirror. According to the law of reflection, the angle of incidence is equal to the angle of reflection. Hence, the light reflects off the mirror at the same \(30^{\circ}\) angle with respect to the normal. This predictable behavior is key to calculating the reflection path.

Reflection Path

Understanding the reflection path is crucial to solving problems in geometric optics. In this exercise, the light travels from the lamp to the mirror and then to the person. This sequence creates the reflected path.

• First, calculate the distance from the lamp to the mirror \((d)\).
• Then, add the distance from the mirror to the person \((d/2)\).

Together, these distances create the reflected path with a total distance of \(\frac{3d}{2}\). The reflected path is often longer than the direct path due to the additional travel segment to the mirror and back. This path is shaped by the angle of incidence, which bounds the trajectory of the light.

Travel Time Ratio

In this exercise, we are interested in finding the travel time ratio between the reflected and direct path.

• The direct path heads straight from the lamp to the person, covering a distance \(d/2\).
• The reflected path takes a detour involving both a journey to the mirror and then to the person, adding up to \(\frac{3d}{2}\).

The travel time along each path is directly related to the distance traveled, assuming the speed of light is constant. Calculate the travel times and form a ratio:\[\frac{\text{Reflected Time}}{\text{Direct Time}} = \frac{\frac{3d}{2c}}{\frac{d}{2c}} = 3\]This ratio of 3 indicates that light takes three times longer to reach the person via the reflected path as opposed to the direct path.

Speed of Light

The speed of light is a crucial factor when calculating travel times for light paths, as it is used to determine how long it takes for light to travel across various distances.

• In this exercise, we use the speed of light, denoted by \(c\), in our formulas to find the time taken.
• The travel time is calculated by dividing the distance by \(c\).

By substituting \(c\) into the equations for both reflected and direct paths:

• Reflected Path Time: \(\frac{3d}{2c}\)
• Direct Path Time: \(\frac{d}{2c}\)

These computations hinge on \(c\), the constant representing the speed of light in a vacuum, approximately \(299,792,458 \text{ m/s}\). Understanding this constant helps us convert distances into tangible time intervals for light travel.