Understanding the Equation of Motion in Particle Mechanics: Explained

[I am Meaningless]

Hello everybody, I Am Meaningless and I had this doubt when I was reading some illustrations based on questions in Particle mechanics.


So the Question was,
A particle is attracted by a force to a fixed point varying inversely as (distance)^n. If the velocity acquired in falling from an infinite distance to a distance 'a' from the centre be equal to the velocity acquired in falling from rest from distance 'a' to a distance (1/4)'a', prove that n = (3/2)
As this was an illustration it was natural for me to directly look at the solution given to me below it.
They started the problem like this:

The Equation of motion is

d^2x/dt^2 = -(u)/x^n
where u = meu (symbol) and x (from my observations) = distance covered

I did not understand this and would love it if someone could explain this equation

EDIT: I recognised the fact that
acceleration (inversely proportional) distance ^n
and when I equated would result in a proportionality constant, taken as 'u' and since it was being attracted had the negative symbol on it. But does the symbol have any significance? Any physical values?

 

[haruspex]

But does the symbol have any significance?

The symbol μ? No, it is just representing an unknown constant.

 

[I am Meaningless]

The symbol μ? No, it is just representing an unknown constant.

Thanks for your help, just was wondering if perhaps it really was an important equation.

 

[haruspex]

Thanks for your help, just was wondering if perhaps it really was an important equation.

No, it represents the behaviour in this artificial question. It does not correspond to any actual physics of which I am aware.

 

[I am Meaningless]

No, it represents the behaviour in this artificial question. It does not correspond to any actual physics of which I am aware.

Okay, Thanks for your help again![emoji1]