Sunday, August 25, 2013

How to solve physics problems

How to think about physics

Before talking about how to solve physics problems, I'd like to talk a little about how to think about Physics.

Physics theories (and their principles and equations) are models of reality. And models are necessarily approximations of realty. That means that they are not 100% accurate -- they are not exactly reality -- at least we haven't found a theory of physics that 100% accurately describes all physical reality. So that raises the question: If our physics theories are only approximations, then why do they work?

Physics theories work because in them we describe constraints where the theories works very well within those constraints, but works very badly for situations outside of those constraints. For example, Newton's laws of motion apply very well to situations where the objects are moving very slow compared to the speed of light, although Newton didn't know that at the time. It was only until 400 years later that Einstein noticed that Newton's laws only work within the constraint of objects moving slow compared to the speed of light.

Further, Einstein explained -- in his book _Relativity: The Special and the General Theory_ -- that Newton's equations of motion are derivations of Einstein's equations. Meaning that if you take Einstein's equations and you constrain the object to much slower than the speed of light, then you get Newton's equations -- it involves doing some limit calculus and algebraic manipulation.

Another source for learning how to understand Physics is the book _The Beginning of Infinity_, by David Deutsch.

Reasoning from first principles

To solve a physic problem (or any kind of problem), one should do it by reasoning from first principles. He should ask the question: What principles matter to this problem? For example, if you have a problem about momentum, then one of the principles to consider is Conservation of Momentum.

In an interview with Kevin Rose, Musk said the following:
I think it's important to reason from first principles rather than by analogy…The normal way we conduct our lives is we reason by analogy… 
We are doing this because it’s like something else that was done..or it is like what other people are doing…slight iterations on a theme… 
“First principles” is a physics way of looking at the world…what that really means is that you boil things down to the most fundamental truths…and then reason up from there…that takes a lot more mental energy… 
Someone could –and people do — say battery packs are really expensive and that’s just the way they will always be because that’s the way they have been in the past…
They would say it’s going to cost, historically it cost $600 KW/hour.  It’s not going to be much better that in the future…
So first principles..we say what are the material constituents of the batteries.  What is the spot market value of the material constituents?  It has carbon, nickel, aluminum, and some polymers for separation, and a steel can..break that down on a material basis, if we bought that on a London Metal Exchange, what would each of these things cost.  oh geez…It’s $80 KW/hour.  Clearly, you need to think of clever ways to take those materials and combine them into the shape of a battery cell, and you can have batteries that are much cheaper than anyone realizes.

A system of solving physics problems

Use the following system loosely. Switch between steps 1 2 and 3 anytime you want, but don't go to step 4 until you pass the scope test -- you should not use principles outside their scope -- and the variable-equation test -- you should NOT have more variables than equations. Then do the two double-checking steps at the end, and if your solution fails either test, then go back through all the steps to look for your error.

Note on #1: Make sure that each object has a different variable. So if you have 2 velocities, then make one of them Vi and the other Vf, or V1 and V2 -- it depends on the context of the specific problem you're trying to solve. This requires your creativity.

Note on #4: One major task is to break up the problem into as many parts as is needed. See examples below.

Vectors -- vectors are used to represent physical quantities that have both magnitude and direction, such as force -- in contrast to scalar quantities, which have no direction. [Learn vector math from the Khan Academy.]
Principles -- things like Newton's Second Law of Motion which says that: The acceleration of a body is directly proportional to, and in the same direction as, the net force acting on the body, and inversely proportional to its mass. Thus, F = ma, where F is the net force acting on the object, m is the mass of the object and a is the acceleration of the object.
Equations -- things like F = ma
Physics-speak -- this is the language of physics
Unknown variable -- this is the variable that the question is asking about
Ballpark figure -- this is the figure you ballparked before using the calculator.


Find the magnitude and direction of the resultant velocity vector for a fish swimming at 3.0 m/s relative to the water across a river facing east that moves at 4.0 m/s. [Hint: The fish-swimming vector (s) and the river-current vector (c) are perpendicular to each other.]

See more examples here.

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