How to Set Up Physics Problems

Physics is a difficult subject.  It can be intimidating, particularly when you have little experience with the subject, as is the case with anyone taking an introductory physics class.  Physics problems will look much less scary, though, once you learn to break them up into small, manageable pieces.  The following is the advice I give to students in my intro classes.

More than 50 years ago, G. Polya described the main parts of solving math problems in his book How to Solve It.  These same steps help with the solutions of physics problem.

1. First, make sure you really understand the problem.  You should be able to paraphrase the problem without adding any new assumptions or removing any of the constraints of the original problem.  Identify all the quantities you are given or asked for and name them with variables.  You should also make a sketch to help you understand (and explain) how things relate to each other geometrically.

2. Next, devise a plan.  This means identifying the equations and principles you will need to solve the problem.  In most cases, you will have to use more than one equation, with the output of one equation becoming the input of another.  Make sure you see these relationships.

3. Once you have reduced the physics problem to a math problem, use your math skills to solve it. Your math prerequisites should be enough for you to do this, but the lectures and textbook will contain important refreshers and explanations for techniques you may not have quite mastered. Even though you are probably not used to doing so, it is important to keep all your units at each step.  This will help you in two main ways.
 a) It makes you think about the numbers you are using.  In introductory physics, we use the acceleration 9.8 m/s² very frequently, but the velocity 9.8 m/s will only appear by chance.  If you see the velocity 9.8 m/s in your calculations, you are probably making a mistake, so double-check.  You won't notice this if you only write 9.8. 
 b) It performs the same check for numerical problems as dimensional analysis does for algebraic problems.

 4. Having carried out the math, the problem is solved, but it is still important to look back at your results and perform any checks you can.  
 a) Sometimes you may see that the answer is obviously wrong.  For example, a test several years ago asked how long it would take a diver off a 10-meter platform to hit the water, and a student answered 300 seconds.  That's five minutes to fall about thirty feet!  An answer like that is obviously wrong.
 b) Check to see whether your answer satisfies the constraints and conditions of the problem. For example, the quadratic equation you must solve for the diver problem above has two solutions:  1.43 seconds before the diver steps off the platform or 1.43 seconds after the diver steps off the platform.  However, the diver did not shoot up through the surface of the water to the level of the platform; before the dive, she was just standing on the platform, not in free fall.  Only the second answer meets the conditions of the problem.
 c) If your problem has a numerical answer, you could make an order-of-magnitude estimate to see if the answer you give is in the right ballpark.  These are most important if you are dealing with a problem involving very large numbers (like are found in problems relating to the planets) or very small numbers (as are found in problems involving atoms or electrons).
 d) If your problem requires an algebraic answer, you could perform dimensional analysis. Dimensional analysis can't tell you that your answer is right, but it can sometimes tell you that your answer is wrong.

The problem-solving sheets will help guide you through these steps.  In practice, the greatest difficulty most students have is with either step 1 or step 2, so let's look at those in more detail.

Understand the Problem

Read through the problem once to get the general idea, but then go back and look for keywords.

Example 1:  Spider-Man falls 10.0 m off a building ledge before a strand of his web starts to slow him. The strand of web brings him to a stop in 2.0 m.  What is the acceleration (assumed constant) as the web catches Spidey?

In this example, there are 3 key positions: the top of the building, where Spider-Man starts falling from rest; the point 10.0 m below that where the web starts to stretch; and the point 2.0 m below that (12.0 m) below the top of the building where he again comes to rest.

Since Spidey is just falling off the building, his initial vertical velocity will be zero when he is at the top of the building.  For the first 10.0 m he is freely falling, so his (vertical) acceleration is a constant -9.8 m/s² over that distance.  For the last 2.0 m he again has a constant acceleration (because both gravity, which continues to exert a force on him, and the tension in the web are  both constant – not a realistic assumption for the web, as a matter of fact), but here the acceleration is not given.  It is what we are looking for.

Notice how useful a sketch would be in keeping this straight in your mind!  Such a sketch does need to show (not necessarily to scale) the two distances through which Spidey falls, but it does not need to contain elaborate artistic details.  A stick figure for Spider-Man is just right!

If sketches are useful for one-dimensional problems, they are indispensable for two-dimensional problems.

Example 2:  A spring-loaded toy cannon shoots a small metal ball off the top of a table with an initial speed of 4.75 m/s.  The ball is fired at an angle of 60^o from the horizon at an initial  height of 1.10 m. How far does the ball move horizontally by the time it hits the floor?

The hint for how we go about this problem comes from the wording of that last sentence.  The solution has two main parts.  (1)  “How far does the ball move horizontally by [a certain] time?”  “How far” is the final result we are looking for, but to get it we need first to find out how much time passes until we stop measuring.  That part comes from the second question:  (2) “At what time does the ball hit the floor?”

As for the sketch(es), in this case you need to show the parabolic trajectory of the ball as it leaves the cannon.  The initial height needs to be shown, as well as the horizontal distance you are seeking and the initial velocity vector.  You should also show (probably in an expanded view) a right triangle showing the initial velocity vector together with its horizontal and vertical components.

Devise a Plan

 1. Write out all the equations you think might be useful in the solution.

 2. Circle every variable that you already know.

 3. Draw a box around the final variable you are looking for. 

 4. If you find one equation in which all the variables are either circled or boxed, you can use that equation to solve the problem.  
 5. If not, see if you have a pair of equations in which 
 a) in one equation, everything is circled but one variable
 b) in the other equation, everything is circled or boxed but one variable, and 
 c) the unmarked variable is the same in both a) and b).  

 6. Draw a parallelogram around the unmarked variable from step 5.  This is your intermediate result.

 7. Find the intermediate result from the equation that has everything circled except for the variable in the parallelogram. 
 8. Use value of the intermediate result in the other equation to find your final result. 

That sounds terribly complicated, but the examples already given (the Spider-Man problem and the projectile problem) should help clarify what I mean. 

Example 1: First some definitions.  Let's call the top of the building point A, the point 10.0 m below that where the web first starts to stretch B, and the point 2.0 m below that where Spidey stops again point C.  Some quantities are defined at these points and will use a subscript A, B, or C; others are defined over the interval AB or the interval BC, so they will use those subscripts.  Both intervals involve constant acceleration, so we can write out our equations for constant acceleration.

1. Δy_AB = v_A t_AB + ½ a_AB t_AB²
2. v_B = v_A + a_AB t_AB
3. v_B² = v_A² + 2 a_AB Δy_AB
4. Δy_BC = v_B t_BC + ½ a_BC t_BC²
5. v_C = v_B + a_BC t_BC
6. v_C² = v_B² + 2 a_BC Δy_BC

The only things we know are 

Δy_AB  = -10.0 m,
v_A = 0 m/s,
a_AB= -9.8 m/s²,
Δy_BC = -2.0 m, and
v_C = 0 m/s,

so draw circles around those variables.  We are looking for a_BC, so draw a box around that. None of the equations have all the variables in circles or boxes, so we follow Step 5 and look for a pair of equations that share one unmarked variable.  Equations 3 and 6 satisfy the requirements, with v_B as the shared unmarked variable.  Draw a parallelogram around it; it is an intermediate result we will need.  Equation 3 now has only circles and the parallelogram; it can be solved for  v_B, then Equation 6 can be solved for a_BC.

Example 2:  To simplify things, I'll assume you have already calculated the x- and y-components of the initial velocity.  In the x-direction you have constant velocity, and in the y-direction you have constant acceleration, so the equations you might use are
1. Δy = v_iy t + ½ a_y t^₂
2. v_fy = v_iy + a_y t
3. v_fy² = v_iy² + 2 a_y Δy
4. Δx = v_ix t 

We know
v_ix = (4.75 m/s) cos 60^o = 2.375  m/s,
v_iy = (4.75 m/s) sin 60^o = 4.1136  m/s,
Δy = -1.1 m, and
a_y = -9.8 m/s²,

so draw circles around those variables.  We are looking for Δx, so draw a box around that.  Once again, when we come to Step 4, there are no equations that have only circled and boxed variables.  However, Equations 1 and 4 are a pair satisfying the requirements of Step 5, with t as the intermediate result, so draw a parallelogram around t.  At this point you should be able to see that you can solve Equation 1 for t and then substitute that intermediate value into Equation 4 to find Δx.

Of course, more complicated problems may require you to chain together three equations connected by two intermediate results, four equations connected by three intermediate results, or whatever.  For example, the problem in Example 1 could have given Spider-Man's mass and asked for the net force on him while the web is slowing him, which would add 

7. F_BC = m a_BC,

making a_BC an intermediate result connecting Equations 6 and 7, just like v_B connects Equations 3 and 6.  In principle you can go through the same steps, starting with circles and boxes, but if you need multiple intermediate results you're probably better off using different-colored pens to keep the different parallelograms separate.  In practice, you will find that, with experience, you can intuitively identify the equations you will need.

Suggestion:  Use the problem-solving sheets to fully work out Examples 1 and 2. 

Why I Am Not a Libertarian

It's not just because many libertarians are, in theory if not in practice, libertines, but rather because we have different views on the appropriate role of government.  To lift a line from the Catholic Encyclopedia, "The goal of the State is the temporal happiness of man...."  However, virtue is a part of temporal happiness, not just eternal happiness (which is the goal of the Church, not the State).  As a result, the State has an obligation to promote virtue, something with which a libertarian would not agree.  That is not to say that the State should be coercive in the promotion of virtue; both justice and prudence place strong limits on what the State should do.

One non-coercive tool for promoting virtue is the "bully pulpit".  Like any tool of government, it can be (and often is) abused, but it is appropriate for there to be some public statements about the character we the people aspire to have.

Anyhow, that is one reason why I am not a libertarian.  Yet at the same time, I would want someone with libertarian habits to always be on hand to ask (to paraphrase the WW2 slogan), "Is this law really necessary?"  Even the most well-intended law may be (or become) too burdensome, intrusive, or punitive.

Why I Can No Longer Call Myself "Conservative"

I cannot call myself "conservative" because that term has come to be loaded with too many connotations that simply do not apply to me.  I realize that there are many people in the same situation who still use the term "conservative", but eventually a break must be made.  A 2-second word followed by a 5-minute clarification just does not make sense.  

Here are a few of the connotations to which I was referring.

1.  Most people think "conservative" means "Republican".  Well, I am not a Republican.  I sometimes vote for Republican candidates, yes, but the only time I voted for a major-party contender in the presidential general election was 1988.  The GOP sometimes calls people with my beliefs part of their "base", but they actually see us a means of getting into political office.  Once in office, their priorities turn out to be rather different from those of their "base".  This has been a problem for many years, but it appears to be getting worse. 




2.  It seems to be expected that a "conservative" favors draconian immigration laws.  This is not the place for a full discussion of immigration, but there does not seem to be any "side" of that discussion with which I can fully agree.  I agree that illegal immigration is, well, illegal, and under most circumstances is also morally problematic.  The same is true of speeding, though; respect for the law is of course important, but so is a sense of perspective regarding the gravity of the offense.  Illegal immigration has been going on so long and in such numbers that any serious effort to deport all illegal aliens would require a police state, and the prospect of becoming a police state is much more frightening than the problem of illegal immigration.

3.  It seems to be expected that a "conservative" will always favor management and oppose unions.  When the huge salaries and bonuses of executives are challenged, the stereotypical conservative will say that (a) their contracts are negotiated, so who are we to question the market?  and (b) contracts are sacred and must be honored.  Where union salaries and benefits are concerned, though, the market suddenly becomes much less infallible and contracts much less sacred.  Somewhere at the root of this is the idea that workers should be grateful for their wages as they would be grateful for a gift -- as though the wealthy and powerful were in fact entitled to the labor of men and women and would be justified in forcing them to work for nothing. 

4.  It seems to be expected that a "conservative" favors an executive branch of ever-increasing strength.  This is usually justified as needed to "get tough on crime" or to "fight terrorism".  The system of checks and balances simply does not work anymore, and several explicit parts of the Constitution are now routinely ignored.  The fact that so much of this was done under the GOP puts the lie to their claim to believe that the original intent of the authors of the Constitution should be the normative interpretation.  Instead, we move closer and closer to being a full-fledged police state.

5.  It seems to be expected that a "conservative" thinks that if anyone in the unfortunate incident was a bad guy, it was Trayvon Martin, and that George Zimmerman is at least manifestly innocent and possibly a hero.  This one has me baffled.  A man with a gun overtakes and confronts an unarmed man who is minding his own business.  The man with the gun is not a policeman or even a security guard; he answers to no one.  The confrontation takes place on a public street.  The confrontation escalates, and the man with the gun kills the man with no gun.  Prima facie, the man who (a) prepared for a fight by bringing the gun, (b) initiated the confrontation, and (c) killed a man has done something wrong.

The only real explanation I can come up with for the "conservative" position is that Obama and Al Sharpton came out strongly against Zimmerman.  For many people, that fact alone means that Zimmerman must be a good guy.

On the other hand, if the prosecution is not able to make its case to the satisfaction of the jury, there should be no second trial in federal court.  We should not allow end runs around the protection against double jeopardy because, once again, it pushes us further in the direction of becoming a police state.  (Do you note a theme?)  So do, of course, attempts by the president of the US and the governor of Florida to influence the outcome of the trial.

..ooOoo..

Of course, it is not right to simply be negative.  If I can't call myself a conservative, I ought to say what I should call myself, and if I don't agree with much of what people associate with conservatives, I should state what principles I do hold.

Having given this some thought, I think I'll call myself a Chestertonian.  I'll have to explain what I mean by that in a separate post.

George H. W. Bush Goes Bald to Support 2-Year Old with Leukemia

If you know me at all, you know how disappointed I am in how much power and pride have accumulated to the Executive Branch.  It is all the more refreshing, then, to see that a former president still has some humility and his priorities in the right order.

Yes, There Are Stupid Questions

We have all heard the proverb, "There is no such thing as a stupid question."  Teachers use this to encourage students to ask the questions they must ask if they are to learn.  As long as the question is an honest question, the proverb is probably true.

Not all questions are honest, though, and dishonest questions are indeed stupid questions that should not be answered. 

Answer not a fool according to his folly, lest thou be made like him. 

-- Proverbs 26:4

I was reminded of this some time ago in an exchange of comments on another blog which had posted something about the FDA making "Plan B" available without a prescription to girls of any age -- no matter how young.  I had stated, in response to some thread, that the pro-life consideration is not the only reason for opposing this action by the FDA.  Someone responded by asking, "If the embryo were not a person, would it still be wrong?"  I refused to play that game.

From the context it was clear we were not discussing horse shoe crab embryos; we were discussing embryonic human beings.  The question was not, "Are human embryos persons?" or "How do you know human embryos are persons?"  Those questions would be legitimate from someone who really wanted to know.  That was not what was being asked, though.  Instead, I was asked to pretend that one group of humans, namely those who are still in the embryonic stage, are not persons.

Not only is this offensive -- exactly as it would be offensive to deny the personhood of any other class of humans -- it renders any further discussion pointless.  If you were to grant me that 1=2, I could show that the US debt is zero, because I would be able show that any number is equal to any other number.  By choosing a particularly bad starting point, you would have destroyed any possibility of a meaningful arithmetic.  In the same way, treating human nature as subject to arbitrary redefinition renders ethics meaningless.

The Biggest Cultural Divide Today

As a follow-up to a long series of comments on one of my earlier posts, I think it's worth going into what I consider the single biggest divide in today's society:  whether or not one agrees that there is some sort of Natural Law.  This divide does not always follow familiar markers.  For example, it might seem that at least all Christians would agree that there must be a law written on the heart (Romans 2:15), but it is not hard to find professed Christians who reject such an idea.  At the same time, it might seem that atheists would not believe this, and certainly many do not, but many deists, agnostics, and outright atheists have always believed in a Natural Law.  This brings up several distinct but related questions.



1.  Are some actions really, objectively wrong?  The question is not, "Are some actions wrong according to the standards of my culture?" nor, "Are some actions wrong in the opinions of most people?" nor any such variation.  Are some actions really wrong, meaning the actor cannot "opt out" of the ethical condemnation?  Is any culture which says otherwise objectively in error?

Many people say No; actions may make us uncomfortable or be disapproved by a society or by the traditions of a culture, but there is no other meaning to "right" or "wrong".  These people would say all rules ultimately are merely agreements we make among each other -- we "play ethics" like we play baseball, so that just like we could change the rules of baseball, nothing prevents us from reformulating our ethical systems.

The answer given here is absolutely fundamental.  If ethics is not a serious subject -- if it is just a game the rules of which we are free to change -- then ethics cannot guide society; when society wants to do something different, it will simply change the rules of ethics.

It should be obvious that I am among those who believe that yes, an objective right and wrong does exist and apply to certain choices.

2.  Assuming we agree that some actions are really, objectively wrong, do they constitute a coherent whole?  If so, what are the basic principles around which these are organized?

At the very beginning of the Nicomachean Ethics, Aristotle says that the objective is happiness, then he notes immediately that people disagree about what "happiness" really means, a disagreement that certainly is still found today.  Is happiness merely a life of pleasure, even if that means being "fat, happy, and stupid" -- even if it means being like the Eloi in H. G. Wells' "The Time Machine"?  Or was Odin right to sacrifice one eye for wisdom?  Does happiness consist primarily in what we experience, or in what we become -- so that for serious wrongdoing "the real and final punishment is having to be the person you are"?

And whose happiness are we talking about?  Is it "every man for himself," or is it the happiness of the whole society?  If it is the happiness of society, does that extend to the very old and the very sick?  Does it extend to criminals?  Does it extend to infants?  The unborn?  What about animals?  Is it OK for a society to utterly crush an innocent person if that would increase the total happiness?  If "it is expedient for us, that one man should die for the people, and that the whole nation perish not," does that make it right?

Although Aristotle's thinking has great merit, I think a better starting point comes from the two greatest commandments.

I.  Thou shalt love the Lord thy God with thy whole heart and with thy whole soul and with thy whole mind.
II.  Thou shalt love thy neighbour as thyself.

The advantages of such a formulation are threefold.  First, ethics is immediately placed in a social context, which avoids the common pitfall of egocentricity.  It is not really clear that "intelligent self-interest" can consistently avoid a tendency towards selfishness.  Secondly, ethics is geared towards our neighbors -- specific people we actually meet -- not towards an airy abstraction like "mankind".  Thirdly, because the First Commandment is oriented towards an eternal, immutable, and omnipresent God, the principles (if not the people studying or implementing them) are not limited to fads or provincialism.  That is, I am not able to make the happiness of my own family, or nation, or race, or political party, or generation into the summum bonum without violating the First Commandment.

3.  Can we know what these basic principles really are?  If so, how?

Of course, the objection could be made that maybe universally applicable moral principles exist, but we cannot be sure what they are.  One may be agnostic about ethics.

One objection is that even if a core of fundamental principles tends to recur in culture after culture, for most of these principles -- perhaps all -- at least one culture can be found that does not hold that principle.  This is no doubt true, but it a statement of anthropology, not of ethics; it is about what people or cultures say, not about what actually is.

The traditional answer, which again I hold, is that we perceive whether actions are right or wrong with our consciences, just as we perceive positions, pressures, temperatures, etc. with our senses.  

Our senses can be dulled -- for example, after spending an hour or more in a dimly lit room, we may not realize how dark it is, or after spending time in a smelly dormitory, we may not notice the smell.  Our senses can also be deceived, especially in complicated situations where our attention is misdirected.  In spite of this, we do not simply give up on our senses altogether, but we use reason to develop a consistent view of the world that is either consistent with our senses or at least provides reasons why our senses are occasionally unreliable.

Our consciences can similarly be dulled (when we habitually ignore our consciences) or deceived (mostly in situations in which multiple principles are in play simultaneously).  The whole project of ethics is to develop a consistent moral worldview that is consistent with our consciences or at least provides reasons why our consciences are occasionally unreliable.

Why I Find Materialism Unconvincing.

It is not uncommon to here people make statements like, "Consciousness and thought are emergent phenomena from the physical processes of the brain.  Consciousness and thought may not be present in the individual neurons, true, but neither is ferromagnetism present in individual atoms.  Bringing a lot of neurons together makes a qualitative difference, just as bringing a lot of iron atoms together does."

I've never found this kind of statement to be at all convincing, and I think I should explain the foremost reason why. 

We start out with no ideas or experience of the world.  First come our sensory experiences:  we see, hear, smell, taste, and feel.  Rather quickly we form ideas about these experiences:  for instance, that the toy does not cease to exist just because we cannot see it any more.  

One of the first divisions we make is between things and people. (We undoubtedly think of plants as like things and animals as like people, at least at first.)  People are in some ways like things; they both persist in existence.  In other ways, though, they are different.  People have intentions; things do not.  People have knowledge; things do not.

When we are a little older, we give names to these.  What behaves like a thing is a material object, what has been called in philosophy for millenia a body.  People are said to have bodies, too, which accounts for why they are in so many ways like things; the differences are accounted for by saying that they have spirits, which have the properties not found in bodies (knowledge, intention, etc.) and lack those properties that are found in bodies (position, size, shape, color, etc.).

From this perspective, it is clear that if anything is to have wrong knowledge -- if anything is to be deceived -- it must be a spirit, not a body or a material object.  Material objects are what is left when we have abstracted away traits like knowledge, so they cannot even have mistaken knowledge.  The idea that "consciousness is an illusion of the body" is just nonsense.

This is, however, precisely the idea put forward by many philosophical materialists.  They believe the universe and everything in it to be bodies in the sense discussed above.  The awkward fact remains that each of us knows the universe only because we have knowledge, and we are aware of the universe only because we have awareness; as soon as we formulate the concepts of spirit and matter, it is first of all clear that we have spirits, and only secondarily clear that we also have bodies.  Without spirits we would not even know the meaning of matter.

What, then, are we to make of the fact that the condition of the brain has such an obvious influence on the ability to think?  First of all, I would say that this is probably analogous to the obvious influence that the condition of the eye has on the ability to see; yet no one would today say that vision is actually in the eye.  Secondly, the approach of Aristotelian philosophy to which we are heirs does not by any means deny the existence of a strong but mysterious connection between the spirit and the body; the soul is the form of the body.

How can anyone think this way in the information age?  Doesn't artificial intelligence prove that knowledge can emerge from carefully arranged material objects?  What about the Turing Test?

Material objects can obviously store and manipulate information.  A library of books is an example; an Egyptian temple wall covered with hieroglyphics is another.  Even without knowing how to read the symbols, we can quantify the information by calculating the Shannon entropy.  Books do not actually contain knowledge, though, because the symbols do not have meaning in themselves.  

The same is actually true about AI.  A machine running an AI program can respond in an appropriate way to external stimuli, but it does so without understanding the meaning of the stimuli or the purpose of the reactions; those are known only to the programmer and the user.  It is really no different than the speaker in a telephone handset; the speaker can transform electronic signals into the sounds of a human voice, but it neither has nor needs any understanding of what is being said.

As for the Turing Test, that really works better as a test for how gullible someone is.  For some people, an 8-ball would pass the Turing Test.

Finally, one might object that the whole problem arises from dividing the world into people and things in the first place.  Maybe things have thought and purpose, too, either in a way we might not understand or as a latent ability -- like a man who is blind because of a problem with his eyes, even though he has no problems with his visual cortex.  If you think that and that different objects remain distinct, you might be an animist; if you think all things together have a collective knowledge and purpose (both perhaps latent), you might be a pantheist.  Many people and many cultures have embraced some variation on this theme.  Such ideas may have no room for the separate existence of spirits, but they are still very different from materialism.