Why do we see the Moon during the day only on certain days and not every day?
The Moon is in orbit around the Earth, and takes 29.5 days to complete its circle. This means that it passes the Sun in the sky once every 29.5 days (called New Moon) and then moves slowly away from the Sun until it's exactly opposite the Sun at Full Moon, two weeks later. At any point in these two weeks, you can see the Sun and the Moon in the afternoon sky at the same time. At Full Moon, the Moon rises in the east at exactly the same time as the Sun sets in the west, and that's the only night in the whole month when the Moon is in the night sky all night long. After Full Moon, the Moon continues in its orbit, moving slowly towards the Sun on the opposite side, and again Sun and Moon can be seen at the same time in the morning sky. After another two weeks, the Moon approaches the Sun closely, and is lost in the Sun's glare for a few days. So, basically, having both the Sun and the Moon in the sky simultaneously is the normal thing, almost every day of the month, while having the Moon only in the night sky happens only one night a month.
The fact that so few people have observed the Moon and Sun in the sky at the same time merely confirms that most people rarely if ever look at the sky.
Contrary to common believe, the moon and sun are completely independent as to which is in the sky at any moment. The Moon moves through the sky every 24 hours, plus it moves approximately 1/29th extra per day (The moon rotates around the Earth every 29 days, thus every day it will move by 1/29th of it's total path). Thus, the moon will appear in the same spot every 24 hours and 50 minutes or so.
So, what does that mean? Simply put, there is a 50% chance that the moon will be somewhere in the sky at any moment in time, and a 50% chance that the sun will be in the sky. Thus, half of the time that the moon is in the sky, it will be day.
Why then, one might ask, is the moon most commonly thought of as visible only at night? There are 2 primary reasons.
I hope all of this helped a bit to explain this. Let me know if I can help you understand further.
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I'll jump on this wagon.
There is a natural occurrence called "Earthshine." This is because the earth will reflect some of the suns light at about 4 times the brightness of the moon at night. This allows the moon to have light hit it and be visible during the day.
Now as for the rotation of them moon, it passes through the sky and will go completely around it's orbit in about 29 days. This means that on some days the moon will be shown during the day (ie: before the waxing First Quarter and after the Waning Second Quarter... in between these phases is the New Moon).
Hope that makes sense!