# It's official, Mechanics is hard

• Beanyboy

#### Beanyboy

So, I've been trying to study Mechanics in my spare time - about an hour or two every day - for over two years now. I'm up at 4:00 am most mornings studying before work. Despite being able to speak French, Italian and Irish fluently, holding a few Masters Degrees, and being a teacher myself, I find it hard - thoroughly enjoyable for the most part, but hard.

I decided to try to unearth why - why am I finding this so difficult and I came across a wonderful book, "Applying Cognitive Science to Education: Thinking and Learning in Scientific and Other Complex Domains" (Reif). In it I read that research on people's understanding of acceleration had shown " that the interpretation of a scientific concept (even one as elementary as acceleration) can be remarkably difficult, even for persons who have been familiar with it for a long time". Apparently, graduate students of Physics and professors of Physics also displayed some misconceptions in understanding.

If you are a learner, struggling with concepts in Kinematics/Dynamics, I hope this helps you feel less stupid. If you are someone who is trying to help those of us who are reaching out for help, please take note.

I disagree.

When I first walk into the first semester of General Physics class, after the formal introduction and explaining how the course will be run, I often ask all my students to stop writing notes, etc. and answer a series of questions verbally based on a number of things that I do:

1. I climb up on a step stool, raise a clump of silly putty high up in the air with my hand, and ask the students "What will happen if I let go of the silly putty?" They all know the answer.

2. I then ask "Do you think the speed that the putty fall is different from the moment I let go of it to the moment right before it hits the floor? If it is different, how different is it?" The majority of the class often gets this right.

Then I release the putty and let it fall to the ground, confirming the students answers.

3. I then climb back down onto the floor, and this time, I tell them that I will toss the putty vertically upwards. I ask them what will happen. Again, the majority of the students are able to answer this.

etc... and then I toss it up in the air.

I do a number of other demos in front of them, always asking them FIRST what they think will happen, and then I do it. The point in all of this, which I tell them after I am done, is that many of the concepts and principles that we will be covering in mechanics are things that they already know!. In fact, these are the things they see practically every single day and are common in their lives! They know what will happen to the putty when I let it go. It is not new. It is not surprising!

What physics does is that we don't just say that the putty will fall to the ground. Instead, we also say HOW it falls to the ground, i.e. how fast does it move at every instant during the fall, how long does it take for it to hit the ground, does it matter how high I had the putty initially? Does it matter if, instead of just letting it go, I actually give it a push downward, ... etc... etc?

So what this means is that mechanics is something that you are familiar with, and something that you can actually test easily! What will be new in a physics class is that we will described these things in mathematical formulation. In doing so, we get a more general idea of the physical concept that governs not only the motion of the putty, but also the building of bridges and the orbits of planets.

So no. Mechanics is not hard, at least, not yet (try solving the equation of motion of a pendulum that has a pivot point wrapped around a spool). General Physics E&M may be hard, but not Mechanics. It is all around you and easy for you to see.

Zz.

The thing that most folks find difficult in mechanics is actually kinematics. This encompasses are more than the usual one dimensional constant acceleration equations. Kinematics in 2D can be quite challenging, and in 3D it can be extremely difficult.

One of the main reasons that people find kinematics difficult is because they tend to ignore it, to simply jump over it. They are often in such a hurry to get to the dynamics aspect of a problem that they fail to carefully set up the kinematics. Given proper attention, kinematics is usually not too difficult and it is the key to most dynamics problems.

Beanyboy
The thing that most folks find difficult in mechanics is actually kinematics. This encompasses are more than the usual one dimensional constant acceleration equations. Kinematics in 2D can be quite challenging, and in 3D it can be extremely difficult.

One of the main reasons that people find kinematics difficult is because they tend to ignore it, to simply jump over it. They are often in such a hurry to get to the dynamics aspect of a problem that they fail to carefully set up the kinematics. Given proper attention, kinematics is usually not too difficult and it is the key to most dynamics problems.
Consideration of the kinematics is particularly important in problems involving multiple pulleys.

Consideration of the kinematics is particularly important in problems involving multiple pulleys.

I could not agree more, but this is only the beginning. I can't tell you how many dynamics problems I have started, only to pause part way through, saying, "I need more kinematics results in order to proceed." It took me years to learn this, but I'm firmly convinced it is true.

Chestermiller
I think we need to put things in a bit of a context here. The "mechanics" that the OP may be referring to isn't any advanced mechanics that we deal in advanced undergraduate level, even. One needs to read another thread created by the OP that may put things into perspective here:

So this is the General Physics-level mechanics we're talking about, not something that invoke Lagrangian-Hamiltonian mechanics. It is why I brought up the exercise/demo that I often do in my class at the beginning of the semester.

Zz.

Apparently, graduate students of Physics and professors of Physics also displayed some misconceptions in understanding.

To me, this is about as absurd as concluding that some chess grandmasters haven't fully figured out how a knight moves!

cnh1995
So, I've been trying to study Mechanics in my spare time <snip>... Despite being able to speak French, Italian and Irish fluently, holding a few Masters Degrees, and being a teacher myself, I find it hard - thoroughly enjoyable for the most part, but hard. Despite being able to speak French, Italian and Irish fluently, holding a few Masters Degrees, and being a teacher myself, I find it hard
Being able to speak another language (or even three others) is not a prerequisite for studying physics, nor is holding one or more Masters' degrees necessarily helpful. I have taught at the college level for close to 25 years, and can state that "being a teacher" isn't necessarily conducive to success in physics, either.

I decided to try to unearth why - why am I finding this so difficult and I came across a wonderful book, "Applying Cognitive Science to Education: Thinking and Learning in Scientific and Other Complex Domains" (Reif). In it I read that research on people's understanding of acceleration had shown " that the interpretation of a scientific concept (even one as elementary as acceleration) can be remarkably difficult, even for persons who have been familiar with it for a long time". Apparently, graduate students of Physics and professors of Physics also displayed some misconceptions in understanding.
I haven't read this book, but without further elaboration, I find Reif's comment difficult to believe. Can you provide an excerpt or two from the book that justifies Reif's conclusion?

I sympathize with the OP in that mechanics is hard because of existing preconceptions formed by faulty observation of Nature and are led to erroneous conclusions.

As @ZapperZ suggests, but before any quantitative work or comparison, throw a piece of chalk straight up in the air and ask
1. What is the acceleration of the chalk at the top of its travel?
Preconception: Zero, because an object has to be moving in order to have acceleration.
2. Other than air resistance, how many forces act on the chalk while it is on its way up?
Preconception: Two, gravity and "the force of the hand".
3. Other than air resistance, how many forces act on the chalk while it's on its way down?
Preconception: One, gravity because "the force of the hand" was depleted by the time the chalk reached maximum height.
4. Ask them to draw the net force on the chalk at different points if thrown on a parabolic trajectory.
Preconception: The force is tangent to the path.

Items 2 - 4 are very common and conform to the Aristotelian view of Nature that you need a force to keep an object moving while the direction of motion is the same as the direction of the force. The reason behind this is understandable. If I push a book gently across a table top, it will move in the direction of the push as long as my hand is pushing it. If I lift my hand, the book will stop moving. Therefore for the book to move it needs to be pushed by my hand (or something else). All this is undeniably true. Here come the erroneous conclusion, therefore any moving object requires a force in the direction of motion.

These preconceptions (as well as others reported in the literature) need to be removed before a serious study of the material can be accomplished, the sooner the better. And yes, one does not have to wait to reach dynamics before mentioning the f word because students are already familiar with it. May the force be with you.

jim mcnamara and PeroK
To me, this is about as absurd as concluding that some chess grandmasters haven't fully figured out how a knight moves!

Schaffer, P.S. and McDermott, L.C. (2005) A research-based approach to improving student understanding of the vector nature of kinematical concepts. American Journal of Physics, 73, 921 - 931

I sympathize with the OP in that mechanics is hard because of existing preconceptions formed by faulty observation of Nature and are led to erroneous conclusions.

As @ZapperZ suggests, but before any quantitative work or comparison, throw a piece of chalk straight up in the air and ask
1. What is the acceleration of the chalk at the top of its travel?
Preconception: Zero, because an object has to be moving in order to have acceleration.
2. Other than air resistance, how many forces act on the chalk while it is on its way up?
Preconception: Two, gravity and "the force of the hand".
3. Other than air resistance, how many forces act on the chalk while it's on its way down?
Preconception: One, gravity because "the force of the hand" was depleted by the time the chalk reached maximum height.
4. Ask them to draw the net force on the chalk at different points if thrown on a parabolic trajectory.
Preconception: The force is tangent to the path.

Items 2 - 4 are very common and conform to the Aristotelian view of Nature that you need a force to keep an object moving while the direction of motion is the same as the direction of the force. The reason behind this is understandable. If I push a book gently across a table top, it will move in the direction of the push as long as my hand is pushing it. If I lift my hand, the book will stop moving. Therefore for the book to move it needs to be pushed by my hand (or something else). All this is undeniably true. Here come the erroneous conclusion, therefore any moving object requires a force in the direction of motion.

These preconceptions (as well as others reported in the literature) need to be removed before a serious study of the material can be accomplished, the sooner the better. And yes, one does not have to wait to reach dynamics before mentioning the f word because students are already familiar with it. May the force be with you.
This is a very considered and very helpful reply. Many thanks. I feel fortunate to have the time to obsess about this, not being pressed for time with "exams looming". That said, I honestly don't know how the average student has time to internalise this, nor how the average instructor has time to cover this demanding material.

Dr.D said:
One of the main reasons that people find kinematics difficult is because they tend to ignore it, to simply jump over it. They are often in such a hurry to get to the dynamics aspect of a problem that they fail to carefully set up the kinematics. Given proper attention, kinematics is usually not too difficult and it is the key to most dynamics problems.

I'm mindful of the sequential nature of the Mechanics material, which is why I'm determined to nail down the Kinematics as best I can in order to be ready to move on.Many thanks for the tips/support.

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I could not agree more, but this is only the beginning. I can't tell you how many dynamics problems I have started, only to pause part way through, saying, "I need more kinematics results in order to proceed." It took me years to learn this, but I'm firmly convinced it is true.
This is very encouraging and wills me push on. Thanks for sharing.

I feel fortunate to have the time to obsess about this, not being pressed for time with "exams looming". That said, I honestly don't know how the average student has time to internalise this, nor how the average instructor has time to cover this demanding material.
A barrier to mechanics is not kinematics per se. It is the ability to translate a physical situation into a set of equations, manipulate these equations algebraically to get a final equation and then translate this equation back into a physical situation. To a lot of people this way of proceeding is unnatural. Because kinematics is usually the first topic where they have to apply it, kinematics is extra difficult until the process becomes a tool.

I have also found that, on average, students have a three week lag between the time they are exposed to the material and the time it becomes "their property". The "looming" exams are a useful prod to bring together what they have mastered up to that point. A common student lament by the time the second hourly exam rolls around is, "I can't believe how I missed the first exam. It was so easy!" That's a sign of learning. So my recommendation to you is to be aware that there is a learning curve and not to wait to understand everything before you move on to the next topic. Things might make more sense in retrospect and you have the luxury to be able to go back anyway.

A barrier to mechanics is not kinematics per se. It is the ability to translate a physical situation into a set of equations, manipulate these equations algebraically to get a final equation and then translate this equation back into a physical situation.

I agree. My earlier comment about fluency in other languages and possession of advance degrees not being prerequisites is related to your point. If the struggling physics student has a weak background in algebra and trigonometry, then he or she will find it difficult to translate the situtation into a set of equations, and even more difficult to solve them.
I have also found that, on average, students have a three week lag between the time they are exposed to the material and the time it becomes "their property". The "looming" exams are a useful prod to bring together what they have mastered up to that point.
I agree. I'm teaching a class in C++ programming this quarter. I spent a lot of time on some of the more difficult points, such as working with pointers, structs, and classes. During class-time I repeatedly asked for questions, but got none, but periodic short quizzes showed that the students didn't "own" these concepts yet. Having these topics on regular quizzes and the lack of success by some of the students gives them the hint that they need to put in some extra effort at mastering them.

Schaffer, P.S. and McDermott, L.C. (2005) A research-based approach to improving student understanding of the vector nature of kinematical concepts. American Journal of Physics, 73, 921 - 931

Where exactly in this paper does it claim that "... graduate students of Physics and professors of Physics also displayed some misconceptions in understanding... "?

1. They NEVER did any survey that involves "professors of Physics". They used General Physics students, TA's, and also pre-college teachers.

2. The survey was asking a more "complicated" scenario of a pendulum. This has more complexities involved than your "tossing a coin" scenario in your other thread.

3. If you truly hold the view that even physics professors (and PhDs in general) have such misconception, then why are you even seeking answers on here, considering that many of us who had helped you, are either "professors of Physics" or physics PhDs?

Zz.

kuruman said:
So my recommendation to you is to be aware that there is a learning curve and not to wait to understand everything before you move on to the next topic. Things might make more sense in retrospect and you have the luxury to be able to go back anyway.

Sage pedagogical advice! I've learned to try to "move on and circle back", and to avail of the retrospective understanding.Incidentally, having looked at American and British system of equations, I found the British SUVAT kinematic equations much easier to cope with. Are you familiar with them?

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Are you familiar with them?

No, what do they say? I find it hard to believe that there is any real difference.

So my recommendation to you is to be aware that there is a learning curve and not to wait to understand everything before you move on to the next topic. Things might make more sense in retrospect and you have the luxury to be able to go back anyway.

Sage pedagogical advice! I've learned to try to "move on and circle back", and to avail of the retrospective understanding.Incidentally, having looked at American and British system of equations, I found the British SUVAT kinematic equations much easier to cope with. Are you familiar with them?
I am familiar with the SUVAT system. It uses 5 equations to say what can be said with only 2. Their up side is that they allow one to answer a kinematic equation numerically in one step by selecting the correct equation that does the job. Their down side is that they muddle the simplicity of the kinematic description by suggesting that all 5 them are equally important. If you start with ##v = u+at## and ##s = ut+\frac{1}{2}at^2##, you can derive all the others using a little algebra to isolate the quantity you are interested in on the left side of the equation. Not only you have 2 equations to remember instead of 5, but also you gain insight into how these equations are put together and where they come form, not to mention sharpening your algebraic skills. All this in turn will build your confidence in your understanding of kinematics and your ability to solve kinematic problems by planning a strategy and following it rather than by figuring out what the correct formula is. Your call.

That's a very good point. I've seen the derivations and, you're right, I really ought to learn how to derive them. I'm no great fan of mindless plug and chug unless working on those type of problems helps to aid some form of deeper understand at the same time. Still, I do think it's a hand mnemonic to get you up and running.

Good call!

I'm no great fan of mindless plug and chug unless working on those type of problems helps to aid some form of deeper understand at the same time.
I taught college mathematics between '79 and '97 and then changed careers for a job in the software industry. At the time I left teaching there was a big push against what was termed rote learning, with the justification that calculators and graphing calculators could do the heavy lifting. Some teachers at the elementary level even went so far as to promote the use of calculators in first grade, presumably so that the precious tykes wouldn't have to trouble their brains with such difficult problems as adding 5 and 8, or multiplying 6 and 7.
I didn't buy that argument, and still don't, as it seemed to me that a certain learned foundation was required. For comparison, musicians and athletes spend countless hours practicing to hone their skills, committing certain moves to "muscle memory." A person who needs to multiply 6.7 by 3.4 and gets 19.34, after inadvertently typing 5.7 for the first number would have no idea that this is wrong if he or she wasn't able to do some mental estimation.

Still, I do think it's a hand mnemonic to get you up and running.
Unless you forget what the letters are supposed to represent. It's pretty obvious the letters v, a, and t represent velocity, accleration, and time, but what about s and u? For a mnemonic to work, you have to know what the symbols in it mean.

@kuruman mentioned that the five equations of SUVAT really boil down to the two equations he wrote. In fact you can boil those two down to a single equation if you know a small amount of calculus.

##s = ut + \frac 1 2 at^2##
Differentiate both sides with respect to t to get ##v = \frac {ds}{dt} = u + at##

Asymptotic
In fact you can boil those two down to a single equation if you know a small amount of calculus.

In my calculus-based General Physics class, I guided the students to derive all the kinematical equations from a=dv/dt (constant a). So even before they make use of any of them, they know where these equations came from, that they are really the "same", and under what condition they are all valid.

As a bonus, when we started to do rotational motion, I made them do the exact, identical exercise with α=dω/dt. (constant α). They immediately see the parallel between linear motion and rotational motion.

Zz.

berkeman
Makes sense. Many things in physics, even basic physics, are not necessarily intuitive given our preconceptions of how the world works. So yes, it make sense it would be somewhat difficult.

Beanyboy
There's also a lot to be said for the geometric view that in a graph of velocity against time, the slope is the acceleration and the signed area under the curve is the displacement.

In that sense, basic constant acceleration kinematics is simple. What may not be simple is the average student's grasp of basic mathematics, algebra and geometry. Or, more generally, the capacity for abstract thought.

Mentor note: Separated the quoted text from the response.
There's also a lot to be said for the geometric view that in a graph of velocity against time, the slope is the acceleration and the signed area under the curve is the displacement.

I agree. I have found that if I try to integrate the Maths, the graphs, and verbal analyses, this leads to a fuller understanding.

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@Beanyboy, when you quote someone, don't type your response between the beginning quote tag ([quote = ...) and the end quote tag ([/quote]). I have fixed at least three of your posts where you have done this, which makes it hard to follow who said what.

Roger and wilco. My apologies.

No, what do they say? I find it hard to believe that there is any real difference.

S = displacement. U = initial velocity. V= final velocity. A = acceleration. T = time. I found that once I learned what the letters meant, it was much easier for me to do the "plug and chug" work. It's a system used extensively in the British educational system and it worked well for me.

S = displacement. U = initial velocity. V= final velocity. A = acceleration. T = time. I found that once I learned what the letters meant, it was much easier for me to do the "plug and chug" work. It's a system used extensively in the British educational system and it worked well for me.

I suppose this works if you are satisfied with a "plug and chug" approach to mechanics. I always found it much more interesting to start with F = m*a, ask myself if a is constant or not, and then proceed to integrate accordingly. I'm not very big on "plug and chug," although I have known hundreds of mediocre students who thought this was the way to go.

I suppose this works if you are satisfied with a "plug and chug" approach to mechanics. I always found it much more interesting to start with F = m*a, ask myself if a is constant or not, and then proceed to integrate accordingly. I'm not very big on "plug and chug," although I have known hundreds of mediocre students who thought this was the way to go.

I like Hewitt's book as it requires you to: learn basic definitions; check your comprehension; consider hand-on applications; apply math skills to solve problems; analyze problems; think and explain problems; evaluate problems; and a bit of "plug and chug" for equation familiarization. Plug and chug is just one tool from the tool-box. I think we'd both agree that a student who solely used "plug and chug" would be a mediocre one.

I suppose this works if you are satisfied with a "plug and chug" approach to mechanics. I always found it much more interesting to start with F = m*a, ask myself if a is constant or not, and then proceed to integrate accordingly. I'm not very big on "plug and chug," although I have known hundreds of mediocre students who thought this was the way to go.

I think of plug and chug as the method whereby you plug numbers into the equations at the earliest opportunity. The course material is then essentially the same few problems over and over again with different numbers. Even something like mass that always cancels out in gravitational problems is always included. Mechanics then becomes an exercise in arithmetic and the physics is lost, or remains hidden in the numbers.

By contrast, a mechanics book such as Kleppner and Kolenkov has almost exclusively more general algebraic problems. And that get you to think about the physics.

Whether you choose to remember all the SUVAT formulas (I find that ##v^2 - u^2 = 2as## is particularly useful) or derive them is more a matter of taste. The good problems require you to manipulate the basic equations in any case.

Beanyboy
Whether you choose to remember all the SUVAT formulas (I find that v2−u2=2asv^2 - u^2 = 2as is particularly useful) or derive them is more a matter of taste. The good problems require you to manipulate the basic equations in any case.

I do not agree that this is simply a matter of taste. As I see it, it is very important on every occasion to work from fundaments that are always true. The "SUVAT formulas" (I hate that name!) imply constant acceleration, but there is a tendency to want to use them without any thought about whether acceleration is constant or not. Most of the problems of the real world do not involve constant acceleration.

Speaking as a learner, I've never been under the assumption that SUVAT implied a single value for acceleration. Never.

It's a mnemonic/approach that's used extensively throughout the British and Irish educational systems. One would imagine that if British and Irish Physics teachers found that it was fundamentally flawed, they'd have ditched it long ago. They are under no obligation whatsoever to use it, but they still do, presumably because they find it helps. The fact that you don't like the name is utterly irrelevant.

Speaking as a learner, I've never been under the assumption that SUVAT implied a single value for acceleration. Never.
This doesn't carry much weight coming from someone who wasn't aware that velocity and acceleration were different concepts, per your thread about tossing a coin -- https://www.physicsforums.com/threads/toss-of-a-coin.949306/

This wikipedia article -- https://en.wikipedia.org/wiki/Equations_of_motion -- distinguishes between constant translational acceleration in a straight line (in which SUVAT is applicable), constant linear acceleration in any direction (with a different set of equations), and constant circular acceleration (also with a different set of equations).
Beanyboy said:
It's a mnemonic/approach that's used extensively throughout the British and Irish educational systems. One would imagine that if British and Irish Physics teachers found that it was fundamentally flawed, they'd have ditched it long ago. They are under no obligation whatsoever to use it, but they still do, presumably because they find it helps. The fact that you don't like the name is utterly irrelevant.
It's not that it's fundamentally flawed -- it's that SUVAT is applicable for some scenarios, such as an object being tossed or falling under the effect of gravity only, but not applicable for more complex situations.

A saying that I heard about 30 years ago seems apropos here:
If the only tool you have is a hammer, everything looks like a nail.

This doesn't carry much weight coming from someone who wasn't aware that velocity and acceleration were different concepts, per your thread about tossing a coin --:

In 1930, Einstein thought he'd found a dent in the Copenhagen interpretation with the "light-box" thought experiment. Bohr pointed out however that he'd overlooked one thing: Einstein's Theory of General Relativity. Similarly, the fact that I've made mistakes when it comes to acceleration and velocity (and no doubt there are more to come) doesn't mean I'm utterly ignorant of all procedural or declarative knowledge of either of these concepts.

At stake here is the pedagogical value of using SUVAT. I've found it useful, others, perhaps, not so. The debate would now be best served if Mark 44, Dr. D and Beanyboy now took a step back and allowed those teachers and learners who have used the SUVAT to pitch in. I'm sure we'd all be interested to hear their thoughts now, since they know ours.