Intuitive explanation of the acceleration vector

In summary, the acceleration vector is defined as the change in velocity over time, and its magnitude is measured in d/s^2. This is a result of the net force acting in a given direction, as stated by the laws of physics. However, the relationship between acceleration and force is not derived mathematically but rather observed through experiments, making it a fundamental aspect of Newtonian physics.
  • #1
Jaysan33
1
0
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
 
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  • #2
Jaysan33 said:
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.
 
  • #3
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).
 

What is the acceleration vector?

The acceleration vector is a mathematical representation of the rate of change of an object's velocity over time. It includes both the magnitude and direction of the change in velocity.

What does a positive/negative acceleration vector mean?

A positive acceleration vector means that an object is speeding up in the direction of the vector, while a negative acceleration vector means it is slowing down in the opposite direction.

How is the acceleration vector related to Newton's second law of motion?

Newton's second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The acceleration vector represents the magnitude and direction of this acceleration.

Can an object have a changing acceleration vector?

Yes, an object's acceleration vector can change if there is a change in the net force acting on it. This could be due to changes in the object's mass, velocity, or direction of motion.

Why is the acceleration vector important in understanding motion?

The acceleration vector is important because it helps us understand how an object's velocity is changing over time. It allows us to predict the future motion of an object and calculate its position, speed, and direction at any given time.

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