Calculating if a projectile from Mars will impact Earth.

[arianabedi]

Homework Statement

This is not a homework assignment, its more of a personal project.

An object (Mass undecided yet) is launched from the North pole of Mars towards the North pole of Sun. It is assumes that Sun, Earth and Mars are in a perfect linear alignment in front of one another and that this assumption remains constant.

With the above said, will the projectile get so much of an influence from Earth's gravity that it goes completely of course or in worst case scenario, impact Earth? Relevant Information:
-From a 3D simulation, using approximate values for diameters of planets (and Sun) and also the average distance of the planets from Sun, I found that the projectile will pass Earth with a distance of 234,803.4 KM , where the strenght of Earth gravity is 0.007 ms^-2 , using Formula¹, mentioned below.

-Both the Mass (and volume) and the speed of this projectile can be adjusted so the safest launch process is provided (Safe: No impacts with planets or an unwanted trajectory

Homework Equations

Formula¹ : $$g_{h}=g_{0}(\frac {r_{e}}{r_{e}+h})^2$$

$$g_{h}$$ Strength of gravity at given height.
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$$g_{0}$$ Strength of gravity at Earth's surface (9.81ms^-2)
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$$r_{e}$$ Earth's mean radius (6,378 KM)
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$$h$$ Given Height from planet's surface.

The Attempt at a Solution

My initial attempt was to simply calculate a downward acceleration but since the acceleration increases with distance I was unable to perform any calculation (Just became a freshman at University).

The second attempt was a visual attempt, by drawing semi accurate 2D map of the scenario and try to work with angles but it proved more difficult than the previous attempt.

4. Finally...
I will need to know how much of a bend will Earth's gravitation influence cause. If knowing the mass and speed of the projectile is critical to the calculations, please inform me so I can figure them out, if not kindly provide me with the necessary formulas so I can input numbers.Regards,
Arian.

 

[arianabedi]

Seems like I can't edit my original post anymore so I am forced to reply to it.

5.Even more info.
After some research, I found a formula which might help: (http://en.wikipedia.org/wiki/Gravitational_acceleration" )
$$\hat{g}=-\frac{GM}{r^2}\hat{r}$$
$$M$$ is the mass of the attracting object,
$$\hat{r}$$is the unit vector from center of mass of the attracting object to the center of mass of the object being accelerated.
$$r$$ is the distance between the two objects.
$$G_{6.673e^-11}$$ is the gravitational constant of the universe.


Unfortunately I am not able to determine a "Unit Vector" from Earth's centre to the centre of mass of the projectile. (Read its http://en.wikipedia.org/wiki/Unit_vector" but its outside of my math knowledge).