RoboTech: November 2021

Q: What is the difference between equation, model, and formula?

An equation usually is a mathematical BALANCE of two operations that are equal. Here are two straightforward examples:

(1) y = 30 - 10 t [this is a math equation with a straight line as its graph]

(2) y = 3 t^3 - 7 (t) (x) + 8 x^3

When an equation REPRESENTS a specific PHYSICAL event, then the equation would become a formula. For instance, #1 above MAY be a FORMULA in the case of uniformly accelerated motion.

Let (y = Velocity at time t);

let (30 represent the INITIAL velocity)

and

let (acceleration be NEGATIVE 10, such as due to gravity).

We can rewrite the equation as a FORMULA: V (t) = V (i) - g t.

Here is another example: I am sure you have learned of the QUADRATIC equation: y = (a x^2) + (b x) + c. Now this is a math equation with a parabola as its graph. We can rewrite this EQUATION and get a FORMULA for a DISTANCE covered by the object in the first example:

s (displacement) = - 5 t^2 + 30 t + 5.

Not ALL equations can be eligible for becoming a formula. Some do.

Some equations MAY represent a SINGULAR event but would NOT be a formula because the event is a special case and not empirical. For example; the following equation: x = 3 t^3 - 2 t^2 + t - 5 may represent the motion of a particle at time t, but it would NOT be an formula of motion for all particles.

Now let’s see what a MODEL is: when architects design a building, they usually construct a MODEL that is to scale of the actual design. This is true for “normal” observable events in our world. But for events that are not readily observable, a MODEL is used to describe how the scientists are imagining/considering the events. In the Newtonian Model, a FORCE is a VECTOR and an UNBALANCED force causes an acceleration. In Bohr’s Model, energy and locales are QUANTIZED. In the wave-particle duality model, light may be a particle or a wave; in Einstein’s Model, the Universe is WARPED (curved).

Bonus: when a new model is accepted by the science world, it has to CORRESPOND to a previously accepted model.

Source:

https://www.quora.com/What-is-the-difference-between-equation-model-and-formula

Q: What is the difference between a model and an equation?

An equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. The value(s) of the variable(s) (or unknown(s)) which satisfy the equality is/are called the solution(s) of the equation.

for e.g.

x^2 -5x + 6 = 0 is an equation. In fact it is a quadratic (order 2) equation in one variable (x) and the solutions (roots) of this equation are x= 2 and x =3

A model is a description of a system using mathematical concepts and language. A model may help to explain a system and to study the effects of different components, and to make predictions about the behaviour of a system in some circumstance. The process developing a model is termed as modelling. A model can take many forms such as statistical models, game theoretic models etc. but in general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.

The traditional models include four major elements

Governing Equations: describe how the values of the unknown (dependent) variables will change w.r.t independent variables for e.g. v = (d/dt)s so velocity is a derivative of displacement w.r.t. time. where v-> velocity, s-> displacement, t->time

Defining Equations: They define new quantities in terms of base quantities for e.g. Red, Blue and Green are defined as primary colors and all other colors may be created taking a certain combination of these three colors

Constitutive Equations: They define relation amongst two physical quantities that is specific to a material or substance for e.g. response of a crystal to an electric field, flow of liquid in a pipe etc.

Constraints: They are the set of one or more predefined conditions which the solution must satisfy

So models are usually composed of relationships (operators - algebraic, functions, differential etc.) and variables (abstractions of system parameters of interest that can be quantified).

There could be various types of models according to their structure such as

Linear vs. nonlinear models

Static vs. dynamic models

Explicit vs. implicit models

Discrete vs. continuous models

Deterministic vs. probabilistic models

Deductive, inductive, or floating models

Physical theories are almost invariably expressed using models.

for e.g. Newtonian laws accurately describe many everyday phenomena, but at certain limits relativity theory and quantum mechanics must be used and even these do not apply to all situations and need further refinement. It is possible to obtain the less accurate models in appropriate limits, for example relativistic mechanics reduces to Newtonian mechanics at speeds much less than the speed of light. Quantum mechanics reduces to classical physics when the quantum numbers are high.

It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases etc. are among the many simplified models used in physics.

The laws of physics are represented with simple equations such as Newton's laws, Maxwell’s equation and Schrodinger equation. These laws are such as a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximate on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. In many other domains like data science models are built based on some assumptions to predict consumer behaviour etc. Since prehistoric times simple models such as maps and diagrams have been used. Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations.

To sum up one may say that a model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, and event occurrence (yes/no).

The actual model is the set of functions that describe the relations between the different variables.

Models can be of various types depending on what they are supposed to do for e.g. Predictive Models, Optimization Models etc.

Model building involves the following primary steps:

Training the model or building the model

Testing the model or Model Evaluation

The complexity of a model may vary from model to model and most of the times there is a trade-off between the accuracy of a model and it’s simplicity.

Some models are explicable others could be a black box. They usually perform with a certain level of accuracy , at times within a specified confidence interval, within the scope (predefined criteria) of that model.

Source:

https://www.quora.com/What-is-the-difference-between-a-model-and-an-equation

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What is the difference between equation, model, and formula?