Problem 8 - College Physics

Question

A woman \(1.7 \mathrm{~m}\) tall stands \(3.0 \mathrm{~m}\) in front of a plane mirror. (a) What is the minimum height the mirror must be to allow the woman to view her complete image from head to foot? Assume that her eyes are \(10 \mathrm{~cm}\) below the top of her head. (b) What would be the required minimum height of the mirror if she were to stand \(5.0 \mathrm{~m}\) away?

Step-by-Step Solution

Answer

The mirror must be 1.6 m tall, regardless of being 3 m or 5 m away.

Step 1: Understand the Problem

The problem involves a woman standing in front of a plane mirror. We need to find the minimum height of the mirror such that she can see her full reflection. We also need to consider her eye level in the calculations.

Step 2: Calculate Eye Level

The woman is 1.7 m tall, with her eyes 10 cm below the top of her head. Her eye level, measured from the floor, is thus:\[ 1.7 ext{ m} - 0.1 ext{ m} = 1.6 ext{ m} \]

Step 3: Relation of Image in Mirror

For the woman to see her full height in the mirror, the line of sight from her eyes to the mirror and back to the feet must make a right angle. Hence, the top of the mirror should be at half the height of her eyes and reach down to half the height of her eyes above the floor.

Step 4: Determine Minimum Mirror Height (3m away)

When standing 3 m away, calculate how much of the mirror is required above and below her eye level:- Start from her eye level (1.6 m):- Half of this height (below eyes to floor): \( 0.8 ext{ m} \).- Half of this height (above eyes in the mirror): \( 0.8 ext{ m} \).Adding these two, the minimum height of the mirror:\[ 0.8 ext{ m} + 0.8 ext{ m} = 1.6 ext{ m} \]

Step 5: Determine Minimum Mirror Height (5m away)

Distance from the mirror doesn’t affect the necessary height of the mirror to see the full reflection. The same calculations apply, meaning the mirror's required height remains:\[ 1.6 ext{ m} \]

Step 6: Summary of Results

The minimum mirror height is dependent solely on the distribution of her reflection above and below her eyes and is therefore consistent regardless of her distance (3 m or 5 m) from the mirror.

Key Concepts

Minimum Mirror Height

When we talk about the minimum mirror height required for a person to see their full reflection, it involves some interesting principles of light and geometry. In our scenario, the woman is 1.7 meters tall, but her eyes are positioned slightly lower, at 1.6 meters above the ground. This difference influences how we calculate the mirror size.

To see her entire image from head to toe, the top edge of the mirror should be at least halfway between her eyes and the top of her head. Simultaneously, the bottom edge should align halfway between her eyes and the floor. This setup ensures the light reflected travels straight into her eyes.

Thus, the minimum mirror height is half the distance from her eyes to the ground plus half the distance from her eyes to the top of her head. Mathematically, it amounts to:

• Halfway from eyes to floor: 0.8 meters.
• Halfway from eyes to the top of head: an additional 0.8 meters.

Adding these up, both components equal 1.6 meters. Therefore, no matter how far she stands from the mirror, the minimum necessary mirror height remains the same.

Geometry of Reflection

The geometry of reflection in plane mirrors plays a crucial role in determining why a person can view their full reflection even when the mirror is shorter than their height.

A plane mirror reflects light at equal angles with respect to the normal (the imaginary line perpendicular to the mirror's surface). This means the angles of incidence and reflection are always equal, following the law of reflection. For a person to view their entire body, their eyes must capture light reflecting from both the top of their head and all the way to their toes.

Therefore, line segments, where light travels and reflects, form a right triangle with the sides these segments trace:

• One from the eyes to the top of the mirror.
• Another from the eyes to the feet.
• The edge of the mirror forms the hypotenuse.

By applying the geometry of reflection, we derive that only half the viewer’s height needs coverage by the mirror, explaining why a shorter mirror suffices.

Distance and Reflection

A fascinating aspect of plane mirror reflections is that the viewer's distance from the mirror does not affect the minimum height of the mirror required to see their full reflection.

When standing different distances, say 3 meters versus 5 meters, from a plane mirror, it might seem intuitive to think the mirror height should change. However, since the image in a plane mirror appears as far behind the mirror as the object is in front, the overall path of light remains the same.

• This creates symmetrical lines of sight.
• The geometry ensures the top of her head and her feet both remain visible within the same frame of the mirror.

Thus, regardless of being 3 meters or 5 meters away, what remains consistent is the geometry requiring merely half of the viewer’s body height as mirror height, regardless of the actual distance they stand from it.