To determine the distance at which an object directly above Earth's equator would not cast a shadow, we need to consider the angular size of the object and its distance from the surface of the Earth.
Let's assume the object is a vertical pole. The shadow it casts will be created by blocking the sunlight, which is coming from a single direction. If the object is far away, the angle at which sunlight hits the object will be nearly parallel, resulting in a smaller shadow. On the other hand, if the object is too close, the angle of sunlight will be more perpendicular, creating a longer shadow.
To calculate the distance at which the object would not cast a shadow, we need to find the distance at which the object's angular size is equal to or smaller than the angular size of the shadow it would cast. This occurs when the shadow's angle is equal to the object's angle.
The angular size of an object is given by the formula:
angular size = actual size / distance
Let's assume the object has a height of h meters. The angular size of the object, α_object, is given by:
α_object = h / distance
The length of the shadow, s, cast by the object is given by:
s = h / tan(θ)
where θ is the angle between the direction of sunlight and the ground. Since the sunlight is coming from directly above, θ is equal to 90 degrees.
Therefore:
s = h / tan(90) = h / ∞ (approaches infinity)
The shadow's length is infinitely long when the object is directly above the surface of the Earth.
To find the distance at which the object does not cast a shadow, we need to set the angular size of the object equal to or smaller than the angular size of the shadow. Mathematically:
α_object ≤ α_shadow
h / distance ≤ ∞
Since ∞ is not a finite value, this inequality cannot be satisfied. Therefore, it is not possible for an object directly above Earth's equator to not cast a shadow regardless of the distance.