[In a previous post, I talked about numerical calculations](http://blog.dotphys.net/2008/10/basics-numerical-calculations/). The basic idea is to use the momentum principle and the following “recipe”:
- Update the position of the particle
- Update the momentum of the particle
- Update the force on the particle
Looks great, right? Well, it mostly is great. I want to give a couple of pointers about the last step, update the force on the particle. How and when can you do this? Really, in numerical calculations, you will see two types of forces:
- Forces that you can calculate: That looks strange, but it’s true. Maybe you are thinking, can’t you calculate all the forces? – the answer is no. Yes, you can calculate the gravitational force and the electromagnetic force. Also, really all forces you are likely to see are one of those two. You can also calculate the force due to a spring(depends on position), the air resistance force (depends on velocity). These types of forces work well in the above numerical recipe.
- Forces that you CAN NOT calculate: These are all the other forces. Typically, these are forces of constraint. Suppose a block slides down a plane. Yes, you can calculate the force the plane exerts on the block, but it depends on things other than just the position of the block. The force the plane exerts on the block is such as to keep the block on the plane. You can not calculate this in the same way as the previous category of forces. Yes, technically the force the plane exerts on the block IS the electromagnetic force. If you want to calculate this force between all the atoms in the two objects, I encourage that.
So what does this all mean? This means that you can not use the above “recipe” for whatever you want. Sorry.
(I have a trick I will show you later)
**Pre Reqs:** [Kinematics](http://blog.dotphys.net/2008/09/basics-kinematics/), [Momentum Principle](http://blog.dotphys.net/2008/10/basics-forces-and-the-momentum-principle/)
What are “numerical calculations”? Why are they in the “basics”? I will give you really brief answer and then a more detailed answer. Numerical calculations (also called many other things – like computational physics) takes a problem and breaks into a WHOLE bunch of smaller easier problems. This is great for computers ([or a whole bunch of 8th graders](http://blog.dotphys.net/2008/09/computational-physics-and-a-group-of-1000-8th-graders/)) because computers don’t mind doing lots of little problems. Why are they “basic”? Well, most text would say they are not basic. I disagree. I think this is a legitimate method for solving problems. In particular, this is a great way of solving problems that can not be solved analytically (meaning solving one hard problem).
**Numerical Calculations are Theoretical Calculations**
Let me just get this out of the way. Numerical calculations and analytical calculations are really in the same “class”. Often people will lump numerical in with “computational experiment” but that is a really bad thing to do. Some others will claim that there are three different “paths” to discover stuff in science: theory, experiment, and simulations. Simulations are the same thing as numerical calculations which are the same as theory. ([I wrote a letter about this in the American Journal of Physics](http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000076000009000797000001&idtype=cvips&gifs=yes))
Let me start with a problem that can be solved analytically. Suppose I have a ball of mass 0.5 kg and I throw this straight up with a speed of 10 m/s. How high will it go?
Continue reading “Basics: Numerical Calculations”
I can’t remember how I found this, but [Scratch](http://scratch.mit.edu) is a graphical programming language developed at MIT. My kids love this. In order to make sure they don’t know more than I do, I created my own scratch program. I am sure someone from the scratch community will attack it for some reason, but I am ok with that.
The program shows a numerical calculation of the motion of a box with a constant force on it. You change the mass and the force. It “sort of” plots the position as a function of time. Don’t worry python, I still think you are the best.
Learn more about this project
I like computers, really I do. Computational physics is a good thing. However, there is a small problem. The problem is that there seems to be a large number of people out there that treat numerical methods and simulations as something different than theoretical calculations. You can tell who these people are because they refer to simulations as “experiments”. But what do these simulations really do in science? What is science really all about?
To me, science is all about models. Making models, testing models, upgrading models. Models. Some examples are the model of gravity. One such model is that there is a gravitational force between any two objects with mass. This force is inversely proportional the square of the distance between them. (This is Newton’s model). Is this model perfect? No. Is this model the truth? No. How did this model come about? Experimental evidence.
Well, how do you make models and what form can they take? To make a model, you collect some observations. The model should agree with these observations. This model could be a physical model (like the globe). It could be a mathematical model (like V=IR). It could be a numerical model – like a [vpython](http://vpython.org) program of a baseball trajectory with air resistance. These are all models.
What does any of this have to do with 8th graders? I claim that any numerical calculation or simulation could be done with a group of 1000 8th graders rather than a computer. What does a computer do? (a computer program really) A program takes a problem and breaks it into a bunch a really small steps. It then does each of these steps and combines them together in some way. Just like a group of 8th graders with TI-89 calculators. Clearly, they are just computing something – they are not a separate type of science (other than theory and experiment).