The Shrinking Shadow Problem
A class maps the Sun's apparent path over one school day in late September. At 9:00 a.m. a meter stick casts a shadow 142 cm long pointing west-northwest. At noon the shadow is only 58 cm long and points nearly due north. At 3:00 p.m. the shadow is 131 cm long and points east-northeast. A student claims, "The Sun must be getting physically closer to us at noon, which is why the shadow shrinks. Then it moves away again in the afternoon." The teacher asks the class to test this claim using only the shadow data and a model of how shadows form.
1. Explain what the shadow length and direction data actually show about the Sun's position in the sky, and evaluate the student's claim that the Sun is physically moving closer at noon.
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The shadow is shortest at noon (58 cm) and longest in early morning and late afternoon (142 cm and 131 cm). A shadow is cast opposite the light source, so the shadow direction tells us where the Sun is: west-northwest shadow means the Sun is east-southeast; north shadow means the Sun is due south; east-northeast shadow means the Sun is west-southwest. Short shadow = Sun high in the sky; long shadow = Sun low near the horizon. So the data show the Sun rising in the east, reaching its highest point in the south around noon, and setting toward the west. The student's claim is not supported. Changing shadow length is explained by the Sun's changing angle above the horizon, not by a change in distance. The Sun stays essentially the same distance from Earth all day; only our viewing angle changes as Earth rotates. The evidence supports an angle explanation, not a distance explanation.
2. If the same class repeated the measurement in late December, predict how the noon shadow would compare to the 58 cm September shadow, and justify your reasoning using the Sun's apparent path.
Read the worked answer
In late December the noon shadow would be longer than 58 cm. In winter at mid-northern latitudes, the Sun's apparent path is lower in the sky, so at its highest point (noon) it is lower than in September. A lower Sun angle produces a longer shadow for the same meter stick. The reasoning chain: season changes the Sun's maximum altitude → lower maximum altitude means a smaller angle between the stick and the incoming sunlight → longer shadow. Note the direction would still be roughly north (Sun due south) at local noon, so the direction stays similar while the length increases. A good answer states the prediction (longer), ties it to lower Sun angle, and keeps the direction reasoning separate from the length reasoning.