# Finding the equations of motion with observations

Let's say that we don't know the equations of motion. I will try to predict where my ball will fall when I shoot it with an angle $$\alpha$$ and and speed $$v$$ by finding the function that describe this.

I will measure every coordinate in the space it has traveled until it has reached the ground. (I will have a finite amount of data point)

If I try to interpolate those data with a polynomial, will I fall on the "last" equation of motion: $$s = s_{0} + v_{0}t+ \frac{1}{2}at^{2} \ \ \ \ \ ?$$

The idea behind that is: We theoretically found those equations of motion and after that with observations and calculations using the set up equations of motion and yes we found that the ball did land on the spot we calculated and so those laws are effectively describing our universe but we didn't prove them, or did we?

What if we make no assumption about the function that will describe my particle movement and just with observed data and interpolation of those, we will find this function. But does this function will equal the theoretically found functions? If yes, does this prove that those equations are indeed true?

## 3 Answers

Fitting observed data to a simple curve can be useful when there is no theory that tells you what the curve should be. But it doesn’t prove anything and may not lead to any real understanding.

On the other hand, in this particular case, finding that the position is quadratic in $$t$$ does suggest a theory: the object is undergoing a constant acceleration.

• But can we define the acceleration as being the third constant of the found equation ? A new definition of it. – Romain B. Jan 16 at 1:46
• No. Acceleration is defined as the second time derivative of position. This is a very general defnition that applies far more widely than your limited scenario. So your curve does not define acceleration; it tells us that acceleration is constant for an object falling a short distance. (But of course we have a theory — gravity — that explains why.) – G. Smith Jan 16 at 1:49

By fitting the data to a curve you are explicitly defining the model you have for motion.

If you start from $$s(t)$$ then you automatically have $$v=\frac{ds}{dt}$$ and $$a=\frac{d^2s}{dt^2}$$.

It doesn't matter what equation you try and start from, you are imposing your model of motion (and acceleration) when you fit to an equation.

The equation you are fitting is the theoretical model you are using (or testing).

What if we make no assumption about the function that will describe my particle movement and just with observed data and interpolation of those, we will find this function. But does this function will equal the theoretically found functions ?

You may find in some circumstances that the equation you fit the data to does a poor job (is a bad fit).

Note in particular the Taylor expansion for a function which says that for any analytical function $$f(x)$$ we can say :

$$f(x) = \sum^\infty_{k=o}\frac{ f^{(k)}{x_0} }{k!}(x-x_0)^k$$

So for any value of $$x_0$$ we can approximate $$f(x)$$ as a polynomial in powers of $$(x-x_0)$$ by neglecting higher terms. In general this will be OK around $$x=x_0$$ but get worse as we move away from $$x_0$$. Some functions have Taylor expansions which are not well behaved like this, but that's a question of mathematical detail.

In practical terms when you fit data to an equation you will use some statistical method like least squares fit, which means you expect differences from a perfect fit and are averaging them out.

If yes, does this prove that those equations are indeed true ?

No.

"True" is not what physics aims for. Accurate and predictability are what we aim for. We want to produce as general a model as possible that is "accurate enough" for our purposes. What is "true" is a philosophical question. We have found the mathematical approach works well, but a successful model in physics produces not just a match for your own data, but can be used by other people to predict the results of their experiments within the required accuracy.

As an example, consider so so-called "Standard Model". It makes vaer accurate predictions and models well the behavior of the world of elementary particles we see. However their are things it does not predict or model at all (e.g. dark matter). So it's a great model, but we suspect their may be an even better model that predicts things slightly differently, but which can't find (yet) and itself may one day be replaced by an even better model.

But "truth" is not the issue : accuracy and predictability are.

If your model was accurate enough for your purposes and could be used to predict the outcome of your experiments to withing your needs for accuracy then it's a good model.

Data taken for this experiment would normally be (x,y) coordinates, as implied in the problem statement. This would indeed yield a quadratic function, such that $$y = f(x^2)$$. Obviously, such a function is NOT a function of time, and since acceleration and velocity are functions of time, you would get no information regarding the vertical acceleration of the object in question.

In order to get acceleration data, you would have to realize that for the given object, the x and y components of displacement are independent of each other, and both the x component of displacement and the y component of displacement are functions of time. Then, measurements of x, y, and t would have to be taken throughout the whole flight path of the object. Once this was done, one equation for x vs. t would have to be developed, and a separate equation for y vs. t would have to be developed. At that point, it would be obvious that x was linear in t, while y was quadratic in t, and the conclusion would immediately be drawn that the object was accelerated in the vertical direction only, assuming that you knew enough physics to know what acceleration is. In no case would you reach the proper conclusions on math knowledge alone, as math can give you an equation of motion, but it cannot tell you what the underlying concept is.