Analytical Prediction

By Hayden McLeod (hmcleod6)

The Main Idea

Analytical prediction uses a mathematical function that can describe the position or velocity of a system at any given time. In contrast to iterative prediction, this means that there is no need to make multiple calculations at small steps in order to find a solution. However, due to the method of derivation of the velocity, the analytical method is only accurate to a high degree when the force applied to the system is constant. Due to this limitation, the iterative prediction method is much more generally applicable.

A Mathematical Model

Unlike the iterative prediction method which is derived directly from the momentum principle, the formula for analytical prediction is based of the formula for the arithmetic mean.

Here, X bar is the arithmetic mean, sigma f x is the sum of all the values, and sigma f is the total number of terms. Since we only need two values of velocity to calculate the average velocity, the formula, for this purpose, can be simplified.

A Computational Model

Below is a visualization of the analytical prediction of the average velocity. As you can see, it takes two points, (a, f(a)) and (b, f(b)), and finds the average between the two points. Also depicted in the image is the incapability to model non-linear curves by comparing the average slope (velocity) compared to the actual slope (velocity).

In addition, an example of using this method in GlowScript can be seen below.

Not only does this code show how to use the analytical prediction method using python/GlowScript, it also shows how you can use that average velocity to calculate the momentum and update the position. These are the typical calculations associated with prediction methods.

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

Connectedness

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History

Historically, it is well known that basic algebra was discovered and implemented well before calculus, especially based on the fact that algebra was used as a basis to develop calculus. For this reason, it can be assumed that the analytical approach to physics problems related to velocity, or any other changing value, was used long before the iterative method. There is no information on who would have used this approach or when this approach was first used in a physics application.

See also

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External links

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