Intuitive explanation of the acceleration vector

[Jaysan33]

Any help on this random question would be greatly appreciated.

The acceleration vector is *defined* as the change in the velocity vector given a change in time, and the magnitude of the acceleration vector is in d/s^2. According to the laws of physics, acceleration in a given direction is the result of a net force in that direction.

For example, in uniform circular motion, centriputal acceleration is constant in magnitude and always directed along the radius of the circle of motion, meaning that there is some constant force directed along the radius.

My question is: How do we know mathematically that the magnitude of the acceleration vector actually represents the acceleration that would result from a given force in that direction?

I know that a lot of the conclusions from physics are the result of experiments, but the acceleration vector seems to be a purely mathemtical operation/definition (i.e. somewhat arbitrary). I'm trying to reconcile the mathematical definition to an actual net force. So, if the centriputal acceleration vector in uniform circular motion had a magnitude of 5 m/s^2 (and assuming a mass of 1 kg), then physics books would say that the net force is 5 Newtons. But this calculation of force seems to be based off a mathemtical definition of an acceleration vector. i.e. how do we know in real life if this would be the actual acceleration as a result of a force of 5 Newtons and vice versa?

Does anyone have an intuitive explanation or anything that can help me out?

Thanks

 

[olivermsun]

My question is: How do we know mathematically that the magnitude of the acceleration vector actually represents the acceleration that would result from a given force in that direction?

We don't "know mathematically." That is the fundamental point of Newtonian physics as a theory to describe what is observed. F = ma.

 

[TheShrike]

Just one thing, in uniform circular motion the acceleration vector is directed towards the centre of the circle, not along the radius. The force is also in this direction.

Your question reduces to "how do we know Newton's #2 is valid?"

We know by experiment; there is no way to derive this relation (unfortunately).