Kinetic Energy?
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Kinetic Energy?
I guess my physics teacher didn't teach me very well...
After looking through some of my old homework while cleaning out my bag for summer, I came across a sheet of equations, all derived by my physics teacher.
According to this sheet, kinetic energy is onehalf mass times velocity squared...but when look at some of the other equations he worked out I found energy was just mass times velocity squared.
I realize I'm looking at two completely different measures of energy, but what's the difference? How'd the one half get stuck into the equation for kinetic energy?
After looking through some of my old homework while cleaning out my bag for summer, I came across a sheet of equations, all derived by my physics teacher.
According to this sheet, kinetic energy is onehalf mass times velocity squared...but when look at some of the other equations he worked out I found energy was just mass times velocity squared.
I realize I'm looking at two completely different measures of energy, but what's the difference? How'd the one half get stuck into the equation for kinetic energy?
 skeptical scientist
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Re: Kinetic Energy?
Energy is defined as the amount of work that can be done in a certain configuration. By conservation of energy, this is the same as the amount of work required to bring about that configuration. So the kinetic energy of an object with mass m and velocity v is the same as the amount of work required to accelerate the object from rest to velocity v.
Mechanical work is force*distance. To accelerate an object to velocity v, you need to impart an acceleration of a for time t=v/a. During this time, the object will travel d=(1/2)at^{2}=(1/2)v^{2}/a units, and to accelerate an object of mass m at rate a, you need to exert of force of F=ma units. So the work done is F*d=(1/2)mv^{2} units.
Mechanical work is force*distance. To accelerate an object to velocity v, you need to impart an acceleration of a for time t=v/a. During this time, the object will travel d=(1/2)at^{2}=(1/2)v^{2}/a units, and to accelerate an object of mass m at rate a, you need to exert of force of F=ma units. So the work done is F*d=(1/2)mv^{2} units.
Last edited by skeptical scientist on Thu Jun 04, 2009 4:48 pm UTC, edited 2 times in total.
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Re: Kinetic Energy?
That...clears up a whole lot. Thanks!
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Re: Kinetic Energy?
Kinetic Energy is only ever ^{1}/_{2} mv^{2}.
You could have
 Circular motion centripetal force involves mv^{2} with no scaling
 Initial derivation on the basis of dimensional consistency (i.e. overall SI units on both sides match up) and it was a consideration of proportionality
 Momentum (p = mv)
 just missed off the 0.5 while jotting it down on that specific occasion?
You could have
 Circular motion centripetal force involves mv^{2} with no scaling
 Initial derivation on the basis of dimensional consistency (i.e. overall SI units on both sides match up) and it was a consideration of proportionality
 Momentum (p = mv)
 just missed off the 0.5 while jotting it down on that specific occasion?
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Re: Kinetic Energy?
The larger questions are
1) Why can we define energy at all?
and
2) Why do we define work to be force*distance?
The answers are related. Forces in physics tend to have this special property that if you move an object along any path, the integral of force*distance (or the scalar product, if force and distance are vector quantities) depends only on the endpoints, and not on the path taken. For this reason, we say that these forces are "conservative". Furthermore, the change in magnitude of the velocity depends only on the value of this integral and the original velocity, and not on the exact details of the motion. This means that we can define things like "potential energy" and "kinetic energy", and find that the total energy is constant  this is called conservation of energy. Once we make this observation, looking at energy gives a very powerful way to simplify equations (such as, for example, letting us turn certain second order differential equations into first order differential equations, which are much easier to solve, and sometimes letting us avoid differential equations entirely).
From the above it's clear that work should be proportional to force times distance. Of course, we could put some constant out front, which might simplify the equation for kinetic energy, but the observation that force*distance=energy was really the fundamental one, so we choose our constant to make the fundamental equation the simplest.
1) Why can we define energy at all?
and
2) Why do we define work to be force*distance?
The answers are related. Forces in physics tend to have this special property that if you move an object along any path, the integral of force*distance (or the scalar product, if force and distance are vector quantities) depends only on the endpoints, and not on the path taken. For this reason, we say that these forces are "conservative". Furthermore, the change in magnitude of the velocity depends only on the value of this integral and the original velocity, and not on the exact details of the motion. This means that we can define things like "potential energy" and "kinetic energy", and find that the total energy is constant  this is called conservation of energy. Once we make this observation, looking at energy gives a very powerful way to simplify equations (such as, for example, letting us turn certain second order differential equations into first order differential equations, which are much easier to solve, and sometimes letting us avoid differential equations entirely).
From the above it's clear that work should be proportional to force times distance. Of course, we could put some constant out front, which might simplify the equation for kinetic energy, but the observation that force*distance=energy was really the fundamental one, so we choose our constant to make the fundamental equation the simplest.
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Re: Kinetic Energy?
Now you're just making my head spin...although that does make sense. Sort of. More than my physics teacher was making.
This was his logic:
Energy = work
work = force * distance
force = mass * acceleration
acceleration = change in velocity / time (but for this, he counted the change in velocity as just being velocity)
So, through this, he said that energy = mass * distance* velocity / time, or energy = mass * velocity ^ 2.
This was his logic:
Energy = work
work = force * distance
force = mass * acceleration
acceleration = change in velocity / time (but for this, he counted the change in velocity as just being velocity)
So, through this, he said that energy = mass * distance* velocity / time, or energy = mass * velocity ^ 2.
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Re: Kinetic Energy?
See, this is why I hate the whole business of high school / no calculus physics. You should be able to do all of physics with a few basic principles and understanding what words like "acceleration", "force", "work", "energy" and so on actually mean. Instead, high school physics classes generally consist of memorizing a whole load of equations, and every time you have a problem you just keep writing down memorized equations until you have enough of them that you can solve for whatever it is you're supposed to solve for.
The other problem with writing down equations and using them to solve without understanding what they mean is sometimes you get the wrong answer, as your physics teacher aptly demonstrated in the above example. His mistake was that velocity=distance/time only works for average velocity, or when velocity is constant. Here, we have an object starting out stationary and reaching a final velocity v, so the equation definitely does not apply.
The other problem with writing down equations and using them to solve without understanding what they mean is sometimes you get the wrong answer, as your physics teacher aptly demonstrated in the above example. His mistake was that velocity=distance/time only works for average velocity, or when velocity is constant. Here, we have an object starting out stationary and reaching a final velocity v, so the equation definitely does not apply.
Last edited by skeptical scientist on Thu Jun 04, 2009 5:20 pm UTC, edited 1 time in total.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
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Re: Kinetic Energy?
That sounds exactly like our physics class...are there many resources out there for learning it right? Or do I just wait a few more years, and hope college straightens it out?
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Re: Kinetic Energy?
In order to learn it right, you really need to learn calculus first, so you'll probably have to wait. You can still get something out of the class, but my caution to you would be to remember not just the equations but also something about what they mean, and how they can go wrong (as in the above example).
P.S. I added a second paragraph to my previous post, about what goes wrong with the "equations first" approach.
P.S. I added a second paragraph to my previous post, about what goes wrong with the "equations first" approach.
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Re: Kinetic Energy?
Thanks for the advice  that actually helps explain a lot of the confusion we experienced in the class.
Guess it wasn't very smart of the school board to hire a biology major as the only physics teacher...
Guess it wasn't very smart of the school board to hire a biology major as the only physics teacher...
 skeptical scientist
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Re: Kinetic Energy?
Sadly, sometimes the school board has no choice since there's a shortage of people who want to be teachers. Maybe all those brilliant mathematicians who failed on Wall Street will go into teaching and add to the supply of good math and physics teachers.
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Re: Kinetic Energy?
skeptical scientist wrote:See, this is why I hate the whole business of high school / no calculus physics. You should be able to do all of physics with a few basic principles and understanding what words like "acceleration", "force", "work", "energy" and so on actually mean.
How bizarre! How can they even justify calling it physics, if there's no calculus? I grumbled a bit in high school because the physics teachers weren't very mathematically rigorous, and they could've coordinated the calculus taught in physics better with what we'd been taught in maths class. But at least all physics students in senior high school did have some (single variable) calculus knowledge.
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Re: Kinetic Energy?
In the US, it's pretty common. A lot more high schools require physics than require calculus, so they are pretty much forced to teach noncalculus physics.
By the way, the lack of mathematical rigor only bothered you in high school? It bothered me all through college. (Is there a single quantum mechanics class anywhere that deals with wave "functions" like the dirac distribution with anything approaching mathematical rigor? I suspect they must exist, but I'd like to see some proof.)
By the way, the lack of mathematical rigor only bothered you in high school? It bothered me all through college. (Is there a single quantum mechanics class anywhere that deals with wave "functions" like the dirac distribution with anything approaching mathematical rigor? I suspect they must exist, but I'd like to see some proof.)
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
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Re: Kinetic Energy?
skeptical scientist wrote:In the US, it's pretty common. A lot more high schools require physics than require calculus, so they are pretty much forced to teach noncalculus physics.
My high school required only up to Geometry (A typical student's math life looks similar to: Algebra 1A, Algebra 1B, Algebra 2, Geometry. Yeah, everyone in my school is fscking retarded). I took physics in 10th grade (honors, so it was full of all the good 11th and 12th graders, so the class didn't suck), and while it was math intensive, I didn't run into a math concept that I didn't know already, except vectors, which I picked up easily enough. I suppose calc's required for the more indepth and higher level physics though.
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Re: Kinetic Energy?
Kow wrote:I suppose calc's required for the more indepth and higher level physics though.
Physics was the reason calculus was invented.
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Re: Kinetic Energy?
I was taught displacement/velocity/acceleration before I was taught calculus, by means of graphs, slopes of graphs, and areas under graphs. Not as good as the real thing, and kind of fails for anything but constant acceleration, but better than just memorising rafts of equations.
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Re: Kinetic Energy?
Alexius wrote:I was taught displacement/velocity/acceleration before I was taught calculus, by means of graphs, slopes of graphs, and areas under graphs. Not as good as the real thing, and kind of fails for anything but constant acceleration, but better than just memorising rafts of equations.
That's how my physics teacher taught kinematic equations. I just went "oh, ok. I guess so." Then we did calculus and next semester he did it again with integration.
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Re: Kinetic Energy?
I think it's universal teaching noncalculus physics in physics classes. Afaik in Australia all the physics classes use prederived equations for everything and don't use any calculus at all. I believe the idea behind it is that students might want to do physics but don't want to do the maths to support it (I.e. your calculus classes in the U.S).skeptical scientist wrote:In the US, it's pretty common. A lot more high schools require physics than require calculus, so they are pretty much forced to teach noncalculus physics.
skeptical scientist wrote:By the way, the lack of mathematical rigor only bothered you in high school? It bothered me all through college. (Is there a single quantum mechanics class anywhere that deals with wave "functions" like the dirac distribution with anything approaching mathematical rigor? I suspect they must exist, but I'd like to see some proof.)
I'll tell you next year when I do quantum mechanics. Unless I switch to maths before then.
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Re: Kinetic Energy?
In the UK i found that the mechanics modules in the A level (that's from ages 16 to 18) maths classes were much better at explaining mechanics than the way we were taught the same stuff in our physics class. In maths, we were taught how to solve ODEs that could be used to represent alot of systems, in physics we were taught the solutions of specific examples (SHM, Pendulums, circular motion) and just told to memorize the equations.
Re: Kinetic Energy?
Kinetec Energy is the extra energy which it possesses due to its motion. The kinetic energy of systems of objects may sometimes not be completely removable by simple choice of reference frame.
Re: Kinetic Energy?
skeptical scientist wrote:By the way, the lack of mathematical rigor only bothered you in high school? It bothered me all through college. (Is there a single quantum mechanics class anywhere that deals with wave "functions" like the dirac distribution with anything approaching mathematical rigor? I suspect they must exist, but I'd like to see some proof.)
They do exist, I had one, but they are rare. My original QM class mentioned the concept, but didn't bother further with it. In fact in that class every function and every distribution had a convergent everywhere defined Taylor expansion. We then had a further class focusing on the mathematical theory of distributions underlying QM the next year.
That said, while it is important to have a vague notion of what is going on in the background, for the very vast majority of theoretical and even mathematical physicists it is unnecessary. That is because almost all of the physics is in the algebra, not in the analysis. The annoying thing is that the free particle and harmonic oscillator are chosen as initial examples for familiarity, and that these also actually require some work to make rigorous. Most of the physics of QM is just as easily captured in finite dimensional systems, spin for example, where all these problems disappear.
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Re: Kinetic Energy?
my physics class doesn't have calculus, and when looking at the kinetic energy equation and momentum just now I noticed that the integral of momentum with respect to velocity is kinetic energy, what is the explanation for this?
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Re: Kinetic Energy?
...At current prices.skeptical scientist wrote:Sadly, sometimes the school board has no choice since there's a shortage of people who want to be teachers.
See, the problem with flat payment for educators is that any skill that helps educating a particular subject, but also is in high demand in industry, isn't factored in. So you get people being paid based on their pure "educational" credentials, not there noneducational credentials (such as their ability to actually do math).
The more useful a particular skill is, the stronger the incentive for people who know the skill not to be in education (at least at the high school level).
At the (research) university level, you have a sort of inversion  your ability to teach is vastly unimportant compared to your ability to research and your mastery of the subject. So you get experts in the field who have little to no skills in teaching.
AM 373 / AM 473 @ University of Waterloo? A pair of quantum mechanics classes taught within the faculty of mathematics, as opposed to Physics. There is a faint hope (I never took them).(Is there a single quantum mechanics class anywhere that deals with wave "functions" like the dirac distribution with anything approaching mathematical rigor? I suspect they must exist, but I'd like to see some proof.)
keeperofdakeys wrote:my physics class doesn't have calculus, and when looking at the kinetic energy equation and momentum just now I noticed that the integral of momentum with respect to velocity is kinetic energy, what is the explanation for this?
I found a decent explanation here:
http://www.physicsforums.com/showthread.php?t=68682
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