Plane Mirrors — Question 4

Question 4

A full-length view needs a mirror only half as tall as the person

A person of height H stands distance d from a vertical plane mirror. The eye is at height h above the floor. Treat top of head and feet as points vertically below or above the eye at the same distance from the mirror.

1. Find the lowest and highest mirror heights needed to see the feet and top of head.

2. Derive the minimum mirror height and evaluate its endpoints for H=1.80 m and h=1.68 m.

Question 4 - Solution

A full-length view needs a mirror only half as tall as the person

(a) Method and calculation

$$y_{\mathrm{bottom}}=(0+h)/2=\boxed{h/2}.$$$$y_{\mathrm{top}}=(H+h)/2\Rightarrow\boxed{L_M=y_{\mathrm{top}}-y_{\mathrm{bottom}}=H/2}.$$

(b) Interpretation and further calculation

$$\boxed{y_{\mathrm{bottom}}=0.84\,\mathrm m,\quad y_{\mathrm{top}}=1.74\,\mathrm m}.$$$$\boxed{L_M=0.90\,\mathrm m,\quad\partial L_M/\partial d=0}.$$

Physical check

The halfway heights follow because the eye and virtual image are equally distant from the mirror plane; moving farther away does not reduce the required mirror length.