To determine which of the options is an inertial frame of reference, let's first understand what an inertial frame of reference is. An inertial frame of reference is a frame in which Newton's laws of motion hold true without any additional forces or accelerations. This means that an object at rest in an inertial frame remains at rest, and an object in motion remains in a constant velocity unless acted upon by an external force.
a) In the case of a child on a moving swing observing the child next to them moving in sync, this is not an inertial frame of reference. The swing is experiencing an acceleration due to the child's movement, making it a non-inertial frame.
b) Similarly, when a child on a moving swing throws a ball to their parents standing on the ground, the swing is still experiencing an acceleration. Therefore, this frame is also non-inertial.
c) On the other hand, when a child observes the motion of the planets in the night sky over many nights, this is a candidate for an inertial frame of reference. The motion of the planets is governed by gravitational forces, and from the observer's perspective, it appears that they are at rest or moving at constant velocities.
d) Finally, when a child observes the motion of the sun across the sky during the day, this is not an inertial frame of reference either. The apparent motion of the sun is due to the rotation of the Earth, which introduces acceleration and non-uniform motion.
Therefore, the correct answer is option C. A child observing the motion of the planets in the night sky over many nights is an example of an inertial frame of reference.