# Motion of an object near a force field

I'm trying to simulate the motion of an object near a force field, given an initial velocity $$v_0$$.

I couldn't find anything that fits what I'm trying to achieve.

I have a point of mass $$m_1$$ in the origin of the Cartesian plane (0.0, 0.0) (blue in the image).

The coordinate system is the Cartesian one.

The position of $$M_1$$ shouldn't change.

I figured out this:

• The attraction force between the blue body and the red body is F defined like this: $$F =G{m_0m_1}/{r^2}$$ by Newton's theory.
• The resulting position of the red body should be the sum of $$v_0$$ and a vector representing the component of the motion directed towards the center.

But I didn't get further than that. How could I calculate the position of the body at time t?

• Are you saying assume the position of $M_1$ is fixed? – Bob D Feb 5 '20 at 14:06
• try to pick one coordinate system: either Cartesian $(x, y)$ or Radial $(r,\theta)$. Plot your FBD and write all motion equations in terms of just one coordinate system. you will get your answer. – Vikash Kumar Feb 5 '20 at 14:59
• @bob d yes sorry I've fixed the question – Fabrizio Feb 5 '20 at 16:30
• @Vikash Kumar what do you mean with fbd? – Fabrizio Feb 5 '20 at 16:32
• Does this answer your question? How to predict the location of a planet at a given time? – Bill N Feb 5 '20 at 17:59

## 1 Answer

For the body with mass $$m_0$$, split the velocity and acceleration vectors into its perpendicular components. Let the body be at position $$(a,b)$$ initially. The displacement due to the initial velocity will be $$s=vt$$. Do the same for the acceleration using the formula $$s=\frac 12at^2$$. Add them together for the x-components and y-components and you arrive at your answer.