Find Velocity, Displacement of Particle w/ Gravity over Time

[mlazos]

we have a particle that is moving along the axe x, then we have the gravity force that is not on the axe x but the center of the gravity (the center of the massive object) has a distance h from the axe x.
In time t what will be the distance that the object will travel?
What are the equations of velocity,displacement with time?
We need the r and h according to time.If i have the equation then i can put different h and r and see the result. Its really important the time, is given.
Particle
*---->-------------------------->x
| Uin (initial Velocity)
|
|
|
|
|
h
|
|
|------------r-----------------* Massive star
_y________________________________________________ ___

Its really important to find the solution , thanks for your time

 

[HallsofIvy]

So the initial velocity is along the x-axis and the "massive star" is at (k, h) in x,y coordinates (I am using "k" for your "r" since I want to reserve r for the distance between the object and the star). I'm going to shift the coordinate system so the star is at (0,0) and the object at (-k,-h). Since you are representing the "massive star" by *, I assume that the distances are great enough that we cannot take the force to be a constant (as we might on the surface of the earth).
The force is given by $F= -\frac{GmM}{r^2}$. The equation you need to use is "mass time acceleration= force":
$$\m\frac{d^2\vec{r}}{dt^2}= -\frac{GmM\vec{r}}{r^3}$$
where $\vec{r}$ is the position vector of the moving object (again, the star is at (0,0)) and r is the distance between the object and the star. (Notice that r³ in the denominator! In order to make that a vector equation with the vector point from the object, we multiply the original gravitational force by the unit vector $\vec{r}{r}$.)
It's not too difficult to solve that for $\vec{v}$, the velocity vector, as a function of $\vec{r}$ but solving for $\vec{r}$ as a function of t is much harder.

This is, of course, the general "planetary orbit" problem and the path is either an ellipse or a parabola.

 

[mlazos]

Thank you for the answer sir, i would like to know if you have any solution for this problem because i tried to find something online but it was impossible.
This problem took me already a week, i thought totday to turn to cylindrical coordinates, do you think this will help?
To express everything in r and theta where theta is the angle of the particle and the massive star.
With given time i need to find what is the maximum distance between the particle and the star so the particle to have time to fall.
Its very tricky and the solution is not obvious.Do you know any webpage or any person i could ask? Someone on this planet had this problem before, i guess NASA too hahaha
thanks for your help and your answer