Challenges of Moon landing: the trajectory confusion

With the modern era of science, development in space technology is growing beyond imagination. But, prospects of space mission is too complicated as it seems.

    

Distance from Earth to Moon is approximately 3.84 lakh km.

Moving from one point to another within Earth's atmosphere is heavily influenced by gravity of Earth i.e 9.8m/s*s,

But moving from Earth to Moon i.e traveling 3.84 lakh km includes various influences:

Non-uniform gravity between Earth & Moon

The gravitational pull between Earth & Moon is not usual as two persons next to each other in space. As defined by Newton's law of universal gravitation that 'every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers'.
 

Two objects under gravity [free fall]

Earth & Moon too attract each other. But at the same time centrifugal force between two exist which tends to separate both away from each other.

Newton's law of universal gravitation working together with centrifugal force

Gravitational pull of other astronomical bodies

Solar system is bounded by many other astronomical bodies i.e planets, sun, meteors & many more. These all to influence the motion of body moving around space.

Point gravitational field around Earth

Solar radiation pressure

In general, Radiation pressure is the pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field.
Solar radiation pressure is due to the sun's radiation at closer distances. All spacecraft experience such a pressure except when they are behind the shadow of a larger orbiting body.

Solar radiation pressure;

on perfect reflector at normal incidence (α=0)

True orbital motion of moon

Approximately Moon moving once around Earth in 27.3 days, its average movement is about 13 degrees per day, or 92 degrees per week. 

Moon moving with average movement of 13 degrees per day

The apparent motion of the Moon from night to night. Each night, it moves about 13 degrees, or about 26 diameters, to the east. 

13 degrees to east per night

Thus the apparent motion of the Moon from hour to hour; each hour it moves about one diameter to the East.

Although the Moon is moving eastward around the Earth, the Earth is also turning to the east and much faster, for it goes all the way around its axis of rotation in just under a day. As a result, although the Moon is moving to the east relative to the stars, the much faster westward motion of the sky is carrying it to the west, so despite its eastward motion relative to the center of the Earth, it rises in the east and sets in the west, just like any other celestial body.

On the average the Moon crosses the sky once every 24 hours and 49 minutes (53 minutes longer than the stellar "day"). As a result, it rises (and sets) later and later every day, until after about 27 days, when it has gone once around the sky relative to the stars, it is back in its original position, rising and setting at its original time(s). 

Moon crosses the sky once every 24 hours and 49 minutes

After all the complex calculation an appropriate time & date is selected for space launch to get the rocket at desired trajectory path.

-By Pravin Maurya