At the origin of an $n$-dimensional space, there exists a single free-moving particle ($\gamma$) with a known mass ($m$) and velocity ($\vec{v}$). There also exists $p$ number of fixed points with known positions. Each point exerts a force on and in the direction of $\gamma$.
$$ ||\vec{f_i}|| = \frac{k_i}{r_i^2} \tag{0} $$
$r_i$ is the distance between $\gamma$ and point$_i$, and $k_i$ is a proportionality constant.
I am trying to model $\gamma$'s position and velocity at any time, $t$. I have made a number of attempts at this problem; however, I am hindered by my somewhat limited mathematical vocabulary. Please, walk me through the steps of this problem, and be clear about your notation. Feel free to begin by solving the problem in its simplest form: where $n = 1$ and $p = 1$.
Example: $n=1$, $p=1$
Here is an example of this problem in its simplest form, where $\gamma$ only moves through one dimension and the force vector is only defined by
$$ f(x) = \frac{k}{(x-x_1)} \tag{1} $$
$x_1$ is the constant position of a given point.
The following image is a graph of the $f(x)$ where $k=10$ and $x_1=-2$.
Obviously, acceleration is given by
$$ a(x) = \frac{k}{m(x-x_1)^2} \tag{2} $$
and velocity is given by
$$ v(x) = \int{a(x)}dt \tag{3} $$
I am going to stop here, because I am not sure I know how to integrate the above equation correctly. Can someone help?