Tricky word problems common in physics?

[Antisthenes]

Just started studying math on my own, and noticed that my high school book in math contains word problems that are tricky because the book gives no instruction on how to solve them. Are these kind of "tricky" word problems common when studying to get a university degree in theoretical physics or astronomy?

I want to get a genuine understanding of quantum physics and cosmology, but I lack any special talent in math, and I really suck at solving tricky word problems. Therefore, it would be nice to know whether mastery of theoretical physics depends on a good ability to solve word problems that are "tricky" (in the sense that one is given almost no prior instruction on how to solve them).

Read the following article which says that many of the brightest high school students fail in STEM programs at college because they have not learned to be creative when solving tricky problems:

https://www.greatschools.org/gk/articles/why-americas-smartest-students-fail-math/

Being able to solve new and daunting problems is important if one wants to become an engineer for example, but is this ability also essential if the "only" goal is to understand the more theoretical (and philosophical) aspects of physics and astronomy?

Have actually become addicted to math, but if it's true that many of the smartest high school students fail at university level math, then I fear that a math noob like me lack the IQ and mental agility to ever understand quantum physics and cosmology, no matter how hard I work. So hope it's not a fact that only those with exceptional minds are able to understand cosmology and theoretical physics.

 

[Vanadium 50]

eing able to solve new and daunting problems is important if one wants to become an engineer for example, but is this ability also essential if the "only" goal is to understand the more theoretical (and philosophical) aspects of physics and astronomy?

I would argue that if you cannot apply something to a situation you haven't seen before, you don't really understand it.

 

[fresh_42]

First of all: What are tricky word problems? I got the feeling that we might mean something different, as your post doesn't sound as if you were talking about group theory, automaton theory or language theory. So here is my first lesson, regardless, which scientific fields you plan to study: Be as precise as possible. If nothing else helps, give an example. Otherwise we could talk for hours without saying anything.

Read the following article which says that many of the brightest high school students fail in STEM programs at college because they have not learned to be creative when solving tricky problems

I won't, because this is nonsense. It neither contains any measurable scale of what many should mean, nor is it right. Even worse, it takes single examples and suggest a general truth, and worst the false conclusion that this would be a necessity.

Being able to solve new and daunting problems is important if one wants to become an engineer for example, but is this ability also essential if the "only" goal is to understand the more theoretical (and philosophical) aspects of physics and astronomy?

This depends at which level you want to understand. If you really want to understand modern physics including quantum mechanics and cosmology, there will be no bypass to a profound understanding of the mathematics used. If you want to write fancy articles in pop science literature, Wikipedia will do, it's wrong anyway what is written there. So the philosophical and physical nature of the universe are two completely different things. And neither has to do with tricky problems. However, a good portion of creativity is needed to solve exercises and find proofs. Other than in school, where often one algorithm to solve one problem is taught, like a 100m run training, and then the next scheme for the next problem and so on, you will not know in science, which tools will work, and sometimes there aren't even tools.

Have actually become addicted to math, but if it's true that many of the smartest high school students fail at university level math, then I fear that a math noob like me lack the IQ and mental agility to ever understand quantum physics and cosmology, no matter how hard I work. So hope it's not a fact that only those with exceptional minds are able to understand cosmology and theoretical physics.

I don't think you should restrict yourself before even started. Stay curious, open minded and diligent - and forget the IQ thing. The world at university will be a different one compared to school, yes, so don't draw conclusions on the unknown, which is my second lesson: do not assume what isn't said.

To summarize it: the ability to solve tricky problems is one property among many which are needed to become a brilliant scientist. However, neither are all scientists little Kopernikus or Gauß, nor is it the most important characteristic. The more you know, the bigger will be your tool box and the easier you will solve tricky questions - not the other way around.

 

[Antisthenes]

@Vanadium 50

I agree, but it's a difference between 1) applying a method to a new situation if you have already been given some instruction on how to apply the method in general, and 2) being asked to solve a word problem without any prior general instruction on how to solve those types of problems.

For example, my high school book explains how to solve equations, and I manage to do that, and enjoy it, but the book also contains words problems with no information on how these problems are related to the equations. No info at all on how to apply the equations. I guess the book assumes that I have a teacher that will explain it, but I learn math on my own, so it's much more tricky to find out what I am supposed to do.

I'm willing to learn all this through trial and error and hard work, but my IQ is pretty much average, and I felt discouraged when reading that even some of the smartest high school students get a shock when discovering that they can't solve math problems at college. Is this because it's a common practice in colleges and universities to just explain an equation in the abstract, on a very theoretical level, but with no information on how to apply the equation in general? If so, then I must admit defeat, which is rather depressing, since I have already invested time and money on learning math, and I actually enjoy it so far, despite my lack of talent, so I want to learn more.

 

[Antisthenes]

@fresh_42

Just noticed your message after having posted my reply to Vanadium. Felt a little better after reading your reply, but the thought of investing 3-4 years of hard work only to discover that I don't have what it takes to understand QM is cause for concern, I guess :)

 

[fresh_42]

@fresh_42

Just noticed your message after having posted my reply to Vanadium. Felt a little better after reading your reply, but the thought of investing 3-4 years of hard work only to discover that I don't have what it takes to understand QM is cause for concern, I guess :)

The highest drop out rate in math is within the first year. I guess physics isn't much different. It is a different world there as in school, the language is different, the structure, the goals. If you want to get an idea beforehand, then have a look at one of the books here: https://openstax.org/subjects. They are free and at the level of late high school - early college.

 

[Antisthenes]

I'm aware of the general difference between university and high school, since I'm at a PhD level in philosophy. Have been creative enough within that field to get entrance to academia without high school. Love being creative within the structures and limits set by philosophy of law, for example. Consequently, one should assume that I would also love tricky mathematical word problems, since they are a combination of creativity and structured thinking, which I'm rather good at within the humanities. But have no confidence in my math abilities since my only experience with it has been failure, until now. The upside is that I now get a dopamine kick each time I discover that I can solve equations that I never managed to understand at school. It's addictive. Can spend 8-12 hours a day on it, since time flies by when being in that state of mind.

On the other hand: have to be down to Earth and admit that I have no talent within this field, and that it might be a waste of time to climb a mountain that is too high. I mean, not everybody can play college football, and I guess the same applies to physics. You must have a "math brain", the right genes, to really understand undergraduate and graduate level physics, I assume. However, if this assumption is wrong, I will continue to spend my days in the timelessness of math. At least I have the nerd brain to enjoy that :)

 

[Antisthenes]

Guess I found the answer to my question here:

"Many physics departments ... offer a course in mathematical physics. This is typically a 1 year, 2-semester course covering a wide range of mathematics that a physics major will need. The purpose of such a course is to give a brief introduction to various areas of mathematics, not from the point of view of rigorous proofs and derivations, but from the point of view of how to use them effectively and correctly, especially when applied to actual physics problems."

https://www.physicsforums.com/insights/mathematical-preparations-2/

This is reassuring :)

 

[Mark44]

For example, my high school book explains how to solve equations, and I manage to do that, and enjoy it, but the book also contains words problems with no information on how these problems are related to the equations.

Well, that's what makes them "word problems." You have to be able to interpret what the text is saying, and write equations that represent those words. That's probably the most difficult part of word problems. Once you arrive at the (correct) equations, things are often pretty straightforward.

 

[fresh_42]

Here are some tips how to tackle those kind of problems:
https://www.physicsforums.com/insights/10-math-tips-save-time-avoid-mistakes/

 

[Antisthenes]

@Mark44

The thing is that my high school book has some chapters where it's explained how equations in that chapter are applied and related to word problems. In those chapters I manage to solve the word problem assignments, because I have been given some prior instruction on how to do it more generally. But then there are chapters which only present the pure math of solving an equation, with no context or explanation of how to apply it. And that's when I have to admit defeat, so far at least.

Therefore I became more optimistic when reading that some physics departments have courses on how to apply abstract math in a particular context. Obviously, it' still very difficult to apply it in new situations, but at least one has a general idea of what an equation is about, and not left totally in the dark.

@fresh_42

Thanks for link, will read it tomorrow, since it's getting late here in Norway :)

 

[Dr. Courtney]

Real Physics means you make your own recipes applying the principles to the problem. If all you can do is apply recipes you learned, you have room for growth.

 

[Vanadium 50]

Two years ago, you posted this thread. You got a lot of good advice, which you rejected. Now you're in a place you don't want to be. Rather than have everyone retype what they wrote last time, I recommend you re-read that thread, and this time follow the advice you got.

 

[Mark44]

I want to get a genuine understanding of quantum physics and cosmology, but I lack any special talent in math, and I really suck at solving tricky word problems.

I guarantee that you won't get far in quantum physics, especially, without a lot more math expertise than you have shown in this and the other thread that Vanadium50 mentioned.

Therefore, it would be nice to know whether mastery of theoretical physics depends on a good ability to solve word problems that are "tricky" (in the sense that one is given almost no prior instruction on how to solve them).

Re-read my response in post #9.

Consequently, one should assume that I would also love tricky mathematical word problems, since they are a combination of creativity and structured thinking, which I'm rather good at within the humanities.

Creativity and structured thinking are important, but a solid foundation in math and physics is crucial. I don't see that creativity and structured thinking in a humanities context is at all relevant here. As an analogy, it's like showing up to a carpentry job with a bag of tools that are used by diamond cutters. The tools you need for understanding quantum physics are attained by years of study of mathematics and physics.

But have no confidence in my math abilities since my only experience with it has been failure, until now.

Well, then, the obvious course of action, which was explained to you in the other thread, is to work on your math abiliities. Also, as explained in the other thread, a cursory study spanning three months is not useful.

 

[Antisthenes]

@Vanadium 50

One of the first rules when interpreting a text, especially during a more or less informal online debate, is not to jump to conclusions. Asking questions and not immediately accepting advice is not the same as rejecting it. Back then I sent micromass a pm, thanking him for his advice. Unfortunately, my math studies got interrupted two years ago, so returning now. Learning from scratch, and learning through problem solving, which actually is fun because one learns a ton of math by solving a good math problem.

One of the main questions in this thread here, however, is whether I actually have enough IQ and mental agility to master a very difficult subject like physics. Because if I struggle hard when solving simple algebraic word problems, despite my ability to understand Kant, Heidegger and pass a 101 university course in logic, then it's tempting to question whether I got the IQ to learn advanced physics.

Have the impression that wanting to learn physics as an adult, after you are 30, with little or no prior math experience, is more difficult and time consuming than being an adult who wants to speak both Chinese and Arabic fluently, with no prior experience of learning a new language. Wanting to really understand QM, with no prior math experience, you not only have to climb Mount Everest, you also have to find this mountain in the middle of the Himalayan range, by first climbing a lot of other mountains just to get there. Which means that you either have to be very talented or have started at an early age to learn what you need. I assume it will take about 10-15 years of full-time work, right? If that is the case, is it realistic to expect from an adult that he or she can learn to master physics after the age of thirty? I mean, how many people at that age have the time and money, or natural talent, to cover such a large canvas?

Math and physics are some of the professions I respect the most, almost as much as I respect ambulance workers and doctors, but I also know when it's wise to question my own abilities and limits. Because not everyone is equally smart in all fields.

 

[Mark44]

One of the main questions in this thread here, however, is whether I actually have enough IQ and mental agility to master a very difficult subject like physics. Because if I struggle hard when solving simple algebraic word problems, despite my ability to understand Kant, Heidegger and pass a 101 university course in logic, then it's tempting to question whether I got the IQ to learn advanced physics.

IMO, being able to understand Kant, Heidegger, etc. isn't very relevant, but a basic understanding in logic is helpful. Struggling while working on algebra problems is probably due more to a lack of experience than a lack of mental horsepower.

Have the impression that wanting to learn physics as an adult, after you are 30, with little or no prior math experience, is more difficult and time consuming than being an adult who wants to speak both Chinese and Arabic fluently, with no prior experience of learning a new language.

I would say that learning physics is easier than learning two unrelated languages, because there is much less memorization required. Learning the vocabulary of a language unrelated to your own native language requires a lot of rote memorization. In contrast, many concepts in mathematics and physics build on previously learned concepts, and follow logically from these earlier concepts.

I assume it will take about 10-15 years of full-time work, right?

Based on what you have written here and elsewhere, I don't think it would take this long. Of course, you would need to get up to speed with algebra, trig, calculus, and elementary physics first, which could possibly be done in two or three years. After that, you would need some differential equations and linear algebra -- I believe these would put you in a good place to start in with quantum physics. I'm not expert in physics, so others might need to correct me here.

 

[Antisthenes]

Thanks for encouraging reply :) I hope you are right that it's mainly lack of experience which is the cause of my struggle with word problems. I mean, when I took the logic course at the university I solved problems on the blackboard in front of the class. I like logic. It's ordered and predictable. Word problems in math appear chaotic in comparison, and that causes brain freeze. Which is probably related to getting angry or impatient feedback when trying to solve math problems as a kid. A scary teacher is really not helpful at that age.

However, it seems like it's necessary to do lab work in order to properly understand physics, as stated by ZapperZ here:

"In fact, I would make the assertion that acquiring such [lab] skills is MORE important for most students in a physics class than knowing the material."

https://www.physicsforums.com/insights/surviving-the-first-year-of-college-2/

That sounds reasonable indeed, but it also proves that you can't really learn QM on your own, as a hobby, right? To properly learn a difficult subject like physics one must study it full time at a university, which is kind of obvious. Fortunately, one can do that for free at Norwegian universities, so have to think about that. Because a superficial understanding of physics is rather pointless.

 

[gmax137]

I think it would be very helpful if you would share one of the problems you find tricky with us. Not to find answers but to show more clearly what you are talking about.

 

[Antisthenes]

The following word problem is in a chapter covering the substitution method, with two linear equations that have two unknown variables, but providing no context or information about how to apply this method.

Tom and Jane are 150 km apart. They plan to meet at a point somewhere between them. Tom uses 3 hours on the journey. He leaves at noon. Jane leaves an hour later than Tom, and she uses 5 hours to travel 150 km. How far away is the meeting point?

Yeah, I know, you are probably laughing now and thinking it's pathetic not being able to solve that problem. That's why I'm questioning my ability to learn physics.

 

[symbolipoint]

The following word problem is in a chapter covering the substitution method, with two linear equations that have two unknown variables, but providing no context or information about how to apply this method.

Tom and Jane are 150 km apart. They plan to meet at a point somewhere between them. Tom uses 3 hours on the journey. He leaves at noon. Jane leaves an hour later than Tom, and she uses 5 hours to travel 150 km. How far away is the meeting point?

Yeah, I know, you are probably laughing now and thinking it's pathetic not being able to solve that problem. That's why I'm questioning my ability to learn physics.

At first reading, it seems to be a fairly common type of constant travel rates problem typically given as an exercise in Introductory Algebra (Algebra 1 ) course. Rate multiplied by time equals distance. Assign variables, maybe set up a table of data, and formulate the needed equations.

 

[Antisthenes]

Thanks for reply :) So I have two questions please:

Is it a common practice in physics to quickly present the pure math of a new equation and then demand that students will figure out by themselves how to use it, by just looking at a word problem?

How does one figure out the speed and assign variables, using the substitution method, when it takes Tom three hours to reach the meeting point and Jane uses five hours to travel 150 km? In other words, I really need to study the very basics of word problems, from scratch, including all the basic math they are based on. Alternatively, my brain is just retarded when it comes to math...

 

[fresh_42]

Tom and Jane are 150 km apart. They plan to meet at a point somewhere between them. Tom uses 3 hours on the journey. He leaves at noon. Jane leaves an hour later than Tom, and she uses 5 hours to travel 150 km. How far away is the meeting point?

1. Read it completely.
2. Read it again, making notes: Tom ## =T##, Jane ##= J##, ##d(T,J)=150\, km##, ##t_{meeeting}= 180\,min##, ##departure\, (T)=1200##, ##departure\,(J)=1300##, ##v(J)=150 km\, /\, 5\, h = 30\, km/h##.
3. Which equations are interesting: ##x(t)=\frac{1}{2}a \cdot t^2+v \cdot t +x_0## with ##a =## acceleration, ##v=## velocity and ##x_0=## starting point for ##x(t) =## location at time ##t##.
4. What do we need? A coordinate system: time - location, say for Jane with the origin at her starting point.
5. What else do we know? No acceleration and zero location at Jane's starting point, so ##x_J(t)=v(J)\cdot t + x_{J,0,}## and ##x_J(60)=0##.

etc.

The last point pretends that Jane started also at ##1200## at some earlier point ##x_{J,0}(0)=-30\,km## as she travels at this speed in an hour ##= 60\,min##. This way we need only one time starting at ##1200 \equiv (t=0)##.

 

[Antisthenes]

Wow, impressed and incredibly grateful for such a quick and thorough reply. Have to take a look at it tommorow, because it's 00:04 here in Norway now. But have to understand this, no matter what.

 

[Mark44]

Tom and Jane are 150 km apart. They plan to meet at a point somewhere between them. Tom uses 3 hours on the journey. He leaves at noon. Jane leaves an hour later than Tom, and she uses 5 hours to travel 150 km. How far away is the meeting point?

How does one figure out the speed and assign variables, using the substitution method, when it takes Tom three hours to reach the meeting point and Jane uses five hours to travel 150 km?

You assign variables to quantities that you don't know.The substitution method that you mention has nothing to do with figuring out speed or assigning variables, and typically is used when you have two or more equations in as many variables. In this case, substitution allows you to arrive at a single equation in one variable.
In this problem, Tom's speed is not known -- only the time it takes him to get to the meeting point. The sentence "she uses 5 hours to travel 150 km" is a bit misleading here, since it's extremely unlikely that she will drive the total distance between her and Tom. What this does give us, though, is her speed, assuming that she travels at a constant speed.

There are two implicit assumptions in this problem -- 1) that both will be traveling at constant speeds, and 2) that the meeting point is at some point between them. For the latter, you can effectively assume that they are traveling along a straight line that joins their starting positions.

The core of this problem is the idea that if your speed is constant, then the distance you travel is the product of your speed times the length of time. As an equation, this is d = r * t.

 

[tnich]

The following word problem is in a chapter covering the substitution method, with two linear equations that have two unknown variables, but providing no context or information about how to apply this method.

Tom and Jane are 150 km apart. They plan to meet at a point somewhere between them. Tom uses 3 hours on the journey. He leaves at noon. Jane leaves an hour later than Tom, and she uses 5 hours to travel 150 km. How far away is the meeting point?

Yeah, I know, you are probably laughing now and thinking it's pathetic not being able to solve that problem. That's why I'm questioning my ability to learn physics.

In post #10 @fresh_42 gives a link to some excellent tips on taking math exams. A lot of these apply to doing homework problems, too. In my opinion one of the most important ones is tip #11 "draw it out". For word problems especially, if you make a sketch of the problem showing (in this case) times, speeds, and distances, and then assign variable names to them, it is much easier to write out the equations you need to solve.

 

[Antisthenes]

Have worked on this for hours now, after seeing that my initial answer was refuted by the math book which says that Tom and Jane meet 112.5 km away from where Tom started. My answer was:

Tom's distance = 30 km/h * 3 hours = 90 km

Jane's distance = 30 km/h * 2 hours = 60 km

Which seems to be right according to my own hand-written graph if Tom leaves at 12:00, Jane starts at 13:00, and both arrive at 15:00, after Tom has traveled 3 hours. 90+60 is the same as the total distance of 150 km. But that answer was wrong. Have not been able to figure out how the meeting point is 112.5 km away from where Tom started.

 

[fresh_42]

I have Tom ##90\,km## and Jane ##60\,km\,,## too, and they both travel at ##30\,\frac{km}{h}## under the assumptions @Mark44 has pointed out. Without them I can't see that there are sufficiently many data.

 

[Antisthenes]

I'm so damn happy that you reached the same conclusion too. Ironically, during a walk outdoors I had quickly reached this conclusion when I started with the problem, because it's common sense, but dismissed it since the substitution method appeared more advanced in my book, and then I tried all kinds of other solutions when reading that the answer is supposed to be 112.5 km. Even checked the book's errata list. By the way, also noticed mistakes in the solution chapter of "Prealgebra" by Openstax.

This has been two days of frustration, and wasted time, kind of. At least I learned math when studying similar word problems in order to figure out a way to reach 112.5. It will take a long time before I forget that number :)

 

[Mark44]

Have worked on this for hours now, after seeing that my initial answer was refuted by the math book which says that Tom and Jane meet 112.5 km away from where Tom started. My answer was:

Tom's distance = 30 km/h * 3 hours = 90 km

Jane's distance = 30 km/h * 2 hours = 60 km

Which seems to be right according to my own hand-written graph if Tom leaves at 12:00, Jane starts at 13:00, and both arrive at 15:00, after Tom has traveled 3 hours. 90+60 is the same as the total distance of 150 km. But that answer was wrong.

I agree with your answer, so I think the book's answer is wrong. You did well to verify that your answer "works" in this problem, a habit that we try to encourage in students. To me, one of the most satisfying things about mathematics, at least at this level, is that after you work a problem, you often can check your answer. For this reason, mathematics is very egalitarian, and doesn't rely on the pronouncement of some authority.
[QUOTE="Antisthenes]Have not been able to figure out how the meeting point is 112.5 km away from where Tom started.[/QUOTE]
I'm surprised that your book has an incorrect answer for this relatively simple problem. Sometimes textbooks in more advanced mathematics occasionally will have typos in their answers, but in my experience, books at lower levels are usually more trustworthy. What book are you using?

 

[fresh_42]

I have tried a meeting point which is not on the direct straight between them, but a triangle, but then there wasn't enough information: Do they meet at the same time? What should Tom's travel time be good for, if not? Whether he travels three hours not knowing his speed tells us nothing if we don't assume that Jane also travels for three, resp. two hours.

 

[Mark44]

By the way, an argument could reasonably be made that Tom and Jane were traveling by bicycle, as it would be difficult (and tedious) to drive 150 km at a constant speed of 30 km/hr.

 

[Mark44]

I have tried a meeting point which is not on the direct straight between them, but a triangle

From the problem statement: "They plan to meet at a point somewhere between them"
It's reasonable to infer that they're both moving along a straight line. If they were each traveling on one side of a triangle, the meeting point wouldn't be between them, at least by the usual meaning of "between."

 

[Antisthenes]

One of the main reasons I like math is that there are no excuses. You can't hide.

The math book is called Sinus 1T. It's probably the most popular high school math book in Norway, used by those who want to study physics or math at the university. The book is used by many high schools. Therefore I find it hard to believe that it contains this kind of error.

Have checked it several times now that it says "112.5 km", and that the assignment number 3.178, in the solution chapter, page 484, is related to the assignment number 3.178 on page 365. (I got narcolepsy, so often triple check things in order to avoid mistakes.)

On the other hand, it might be relevant indeed to mention that the original text is actually ambiguous in Norwegian, since it says that it takes Tom three hours to drive "the whole stretch", which in Norwegian often refers to the particular distance a person intends to travel, but it can also refer to the whole stretch of 150 km.

Upon reflection I assume that the book doesn't contain an error, and that I jumped to conclusions when reading the sentence "the whole stretch", right?

 

[martinh]

One of the main reasons I like math is that there are no excuses. You can't hide.

The math book is called Sinus 1T. It's probably the most popular high school math book in Norway, used by those who want to study physics or math at the university. The book is used by many high schools. Therefore I find it hard to believe that it contains this kind of error.

Have checked it several times now that it says "112.5 km", and that the assignment number 3.178, in the solution chapter, page 484, is related to the assignment number 3.178 on page 365. (I got narcolepsy, so often triple check things in order to avoid mistakes.)

On the other hand, it might be relevant indeed to mention that the original text is actually ambiguous in Norwegian, since it says that it takes Tom three hours to drive "the whole stretch", which in Norwegian often refers to the particular distance a person intends to travel, but it can also refer to the whole stretch of 150 km.

Upon reflection I assume that the book doesn't contain an error, and that I jumped to conclusions when reading the sentence "the whole stretch", right?

Tom is traveling at 50 km/h and Jane at 30 km/h. Tom starts an hour earlier, so travels for (1+t) hours compared to t hours for Jane.

So,

50 (1+t) + 30 t = 150

Thus, t = 1.25 hours

Distance traveled by Tom is 50 (1+t) = 112.5 km.

The book is correct.

 

[tnich]

Tom is traveling at 50 km/h and Jane at 30 km/h. Tom starts an hour earlier, so travels for (1+t) hours compared to t hours for Jane.

So,

50 (1+t) + 30 t = 150

Thus, t = 1.25 hours

Distance traveled by Tom is 50 (1+t) = 112.5 km.

The book is correct.

However, that would make Tom's travel time 2.25 hrs. The problem states that his travel time is 3 hrs. How did you decide that his speed is 50km/hr?