The walls of a prison cell are perpendicular to the four...

The walls of a prison cell are perpendicular to the four cardinal compass directions. On the first day of spring, light from the rising Sun enters a rectangular window in the eastern wall. The light traverses $2.37 \mathrm{~m}$ horizontally to shine perpendicularly on the wall opposite the window. A young prisoner observes the patch of light moving across this western wall and for the first time forms his own understanding of the rotation of the Earth. (a) With what speed does the illuminated rectangle move? (b) The prisoner holds a small square mirror flat against the wall at one corner of the rectangle of light. The mirror reflects light back to a spot on the eastern wall close beside the window. How fast does the smaller square of light move across that wall? (c) Seen from a latitude of $40.0^{\circ}$ north, the rising Sun moves through the sky along a line making a $50.0^{\circ}$ angle with the southeastern horizon. In what direction does the rectangular patch of light on the western wall of the prisoner's cell move? (d) In what direction does the smaller square of light on the eastern wall move?

The walls of a prison cell are perpendicular to the four cardinal compass directions. On the first day of spring, light from the rising Sun enters a rectangular window in the eastern wall. The light traverses $2.37 \mathrm{~m}$ horizontally to shine perpendicularly on the wall opposite the window. A young prisoner observes the patch of light moving across this western wall and for the first time forms his own understanding of the rotation of the Earth.
(a) With what speed does the illuminated rectangle move? (b) The prisoner holds a small square mirror flat against the wall at one corner of the rectangle of light. The mirror reflects light back to a spot on the eastern wall close beside the window. How fast does the smaller square of light move across that wall? (c) Seen from a latitude of $40.0^{\circ}$ north, the rising Sun moves through the sky along a line making a $50.0^{\circ}$ angle with the southeastern horizon. In what direction does the rectangular patch of light on the western wall of the prisoner's cell move? (d) In what direction does the smaller square of light on the eastern wall move?

Key Concepts

Apparent motion of light spots

The movement of a patch of light on a surface is a geometric projection effect, not the transport of matter or energy. This means that while the illuminated region can appear to move very quickly—even faster than the speed of light—this does not violate relativity, as no signal or matter is traveling at that speed.

Earth’s rotation and angular motion

The Earth rotates at a constant angular velocity, which leads to a gradual change in the angles at which sunlight strikes a stationary object or surface. This rotation converts into an apparent horizontal displacement of sunlight on surfaces, linking the observed speed of the light patch to Earth’s rotational motion.

Projection geometry

Understanding the displacement and speed of the light patch involves principles of projection geometry. The orientation of the light source, the window, and the walls (which are aligned with the cardinal directions) requires the use of trigonometric relationships to relate the angular movement of the Sun to the linear movement of the patch on the wall.

Reflection laws and mirror effects

When a mirror is introduced, the law of reflection (the angle of incidence equals the angle of reflection) determines how light is redirected. This introduces additional kinematic considerations, as the movement of the reflected spot on another surface is governed by both the mirror’s geometry and the incident light’s path.

Transcript

Please give Ace some feedback