Analysis vs arithmetic approach to solving motion

[rudransh verma]

https://www.feynmanlectures.caltech.edu/I_10.html
https://www.feynmanlectures.caltech.edu/I_09.html
Using Mathematical approach we can describe the motion of a falling body whose gravity is 32 m/s^2. Analysis shows that this is simply $s-s_0=ut+1/2at^2$. Similarly we can describe the motion of a mass attached to spring by mathematical approach. Analysis shows this is simply $x=\cos t$. How do we reach to this conclusion?

 

[Chestermiller]

https://www.feynmanlectures.caltech.edu/I_10.html
https://www.feynmanlectures.caltech.edu/I_09.html
Using Mathematical approach we can describe the motion of a falling body whose gravity is 32 m/s^2.

9.8 m/s^2

 

[rudransh verma]

9.8 m/s^2

Oh yes! acceleration is 32.

 

[Herman Trivilino]

Using Mathematical approach we can describe the motion of a falling body whose gravity is 32 m/s^2. Analysis shows that this is simply s−s0=ut+1/2at2. Similarly we can describe the motion of a mass attached to spring by mathematical approach. Analysis shows this is simply x=cos⁡t.

This is called modelling, a process by which we create models.

How do we reach to this conclusion?

By comparing our model to observations of how Nature behaves. If the model doesn't match what we observe, we try a different model. It's this process of trial and error that leads us to these "conclusions".

 

[nasu]

Oh yes! acceleration is 32.

You've got the units mixed up. This is in ft/s².

 

[rudransh verma]

By comparing our model to observations of how Nature behaves. If the model doesn't match what we observe, we try a different model. It's this process of trial and error that leads us to these "conclusions".

So you are saying we can't derive mathematically the model or analysis from the numerical process. Is it a guess that $x=\cos t$ perfectly describes the motion of spring?

 

[jbriggs444]

So you are saying we can't derive mathematically the model or analysis from the numerical process. Is it a guess that $x=\cos t$ perfectly describes the motion of spring?

We guess and confirm Newton's second law that $F=ma$
We guess and confirm Hooke's law that $F=kx$
We solve the resulting differential equation and obtain a family of solutions that includes $x=\cos t$ and phase-shifted and scaled versions thereof.

 

[Herman Trivilino]

So you are saying we can't derive mathematically the model or analysis from the numerical process.

You can derive the mathematical model and you can perform a numerical analysis using the mathematical model. What you can't do is use that model to reliably predict how Nature will behave. You have to perform experiments to see if your model correctly predicts how Nature behaves.

It was once thought that the power of the human intellect would be all that's necessary, but we have learned in recent centuries that that alone won't tell us how Nature behaves. We need experiments for that.

Is it a guess that x=cos⁡t perfectly describes the motion of spring?

Well, it doesn't perfectly describe the motion of a bob hanging from a spring. It's an idealization call simple harmonic motion, and although we can't produce perfect simple harmonic motion, we can go through a process where we get closer and closer to the idealization. In fact, there is nothing we know of that is a perfectly valid theory of physics. All theories have limits of validity.

As to whether or not it was a guess, I don't know as I'm not aware of the history. But the process of constructing physical theories is an inductive one. We can't deduce the correct theories of physics using deductive reasoning alone.

In mathematics we use proofs that involve pure deductive reasoning. The problem with that process is that there's nothing introduced that wasn't already present in the premises.

Take Einstein's special theory of relativity. He used two postulates to deduce effects like relative simultaneity, length contraction, and time dilation. These deductions follow logically from the postulates using pure deductive reasoning. However, those deductions had to be tested experimentally before they were accepted.

 

[Baluncore]

Using Mathematical approach we can describe the motion of a falling body whose gravity is 32 m/s^2. Analysis shows that this is simply $s-s_0=ut+1/2at^2$. Similarly we can describe the motion of a mass attached to spring by mathematical approach. Analysis shows this is simply $x=\cos t$. How do we reach to this conclusion?

The simple equations are the solution of the differential equations for the model.

 

[rudransh verma]

You can derive the mathematical model and you can perform a numerical analysis using the mathematical model. What you can't do is use that model to reliably predict how Nature will behave. You have to perform experiments to see if your model correctly predicts how Nature behaves.Well, it doesn't perfectly describe the motion of a bob hanging from a spring. It's an idealization call simple harmonic motion, and although we can't produce perfect simple harmonic motion, we can go through a process where we get closer and closer to the idealization. In fact, there is nothing we know of that is a perfectly valid theory of physics. All theories have limits of validity.

As to whether or not it was a guess, I don't know as I'm not aware of the history. But the process of constructing physical theories is an inductive one. We can't deduce the correct theories of physics using deductive reasoning alone.

In mathematics we use proofs that involve pure deductive reasoning. The problem with that process is that there's nothing introduced that wasn't already present in the premises.

Take Einstein's special theory of relativity. He used two postulates to deduce effects like relative simultaneity, length contraction, and time dilation. These deductions follow logically from the postulates using pure deductive reasoning. However, those deductions had to be tested experimentally before they were accepted.

https://www.feynmanlectures.caltech.edu/I_01.html "The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific “truth.” But what is the source of knowledge? Where do the laws that are to be tested come from? Experiment, itself, helps to produce these laws, in the sense that it gives us hints. But also needed is imagination to create from these hints the great generalizations—to guess at the wonderful, simple, but very strange patterns beneath them all, and then to experiment to check again whether we have made the right guess. This imagining process is so difficult that there is a division of labor in physics: there are theoretical physicists who imagine, deduce, and guess at new laws, but do not experiment; and then there are experimental physicists who experiment, imagine, deduce, and guess."

This is the paragraph from the link. I think this is what you are saying.
All the laws like coulombs law, law of gravity, gauss law, kinematic eqns, friction law, hookes law are deduced together from like mathematically/analytically and experiments. They are approximate. These models or laws describe our world roughly exactly.
Things in physics move forward from defining ,experimenting, deducing, experimenting.