Here an example of a question that would explain it:
$a$ is acceleration, $v$ is velocity, $r$ is the distance between the initial position of the object and it's current position.
There is an object (represented by the gray square) that is being attracted to a point (represented by the purple circle) by a force. The point doesn't have a mass and cannot be affected by forces (its just theoretical). This force causes an acceleration on the object. In the situation on the left there is a constant acceleration towards the the purple circle on the gray square, say $a = 1$
If $a = 1$, then $v = \int a = at = t$
If $v = t$, then $r = \int v = \frac{1}{2} at^2 = \frac{1}{2} t^2$
So, $r = \frac{1}{2} at^2 = \frac{1}{2} t^2$
Okay, so that's simple. Now, what if there is a situation where acceleration is dependent on $r$. If you take a look at the right side of the diagram, $r_{gp}$ is the distance between the gray and purple objects and $r_i$ is the initial distance between the objects. I forgot to mention that the maroon square is the initial position of the square object. In this situation, $a$ is going to be dependent on $r_{gp}$. They are going to be inversely proportional. So, $a = \frac{1}{r_{gp}}$.
Because $a = \frac{1}{r_{gp}}$ and $r_{gp} = r_i – r$, $a = \frac{1}{r_i – r}$
This is all good, but when you try too figure out an equation for $a$ with respect to $t$, time, it becomes very tricky because $r$, as seen above, is calculated by integrating acceleration with respect to time and not position.
My question is, is it possible to calculate an equation for $a$ with respect to $t$, and if so, what would that equation be?
Thanks!
Best Answer
You can try using $a=\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=v\frac{dv}{dx}=f(x)\Rightarrow \int vdv=\int f(x)dx$ (I have used chain rule above. Alternatively if you're in a system where only conservative forces are acting, you can write$F=\frac{\partial V}{\partial x}\Rightarrow \frac{1}{m} \frac{\partial V}{\partial x}=f(x)\Rightarrow \int dV=m\int f(x)dx$ (for the part where you can treat partial as normal derivative) where V is potential,m is mass and F is force.