The Bridge Between Maths and Physics
*This thread is purely for people who chose both Maths and Physics*
In almost every physics lesson, our class is reassured that this topic 'isn't going to be mathsy', you know, because that's bad right? Physics and Maths are two separate subjects aren't they? They shouldn't be connected in any way whatsoever!? RIGHT?!
WRONG!
I'm pretty 'sad' when it comes to physics -- from others' 'frame of reference' [giggles]. I read a lot about it in my free time and mess about with simple concepts, trying to figure out how they work. In that time, I 'discovered' a few things that we weren't taught that I just think NEEDS to be brought up. These things have greatly increased my understanding of what physics is to me.
In today's lesson, I hope to kick things off with how integration and differentiation are entwined with SUVAT equations!
You know suvat equations right? The boring equations used to calculate the displacement of a particle, its velocity and who knows what other useless things...right? If you seriously think that, then get ready for your eyes to be...opened [explosion].
In all cases in physics AS, we take the acceleration due to gravity to be constant at 'g'. Fundamentally this is wrong, it changes with respect to the distance from the massive [scientifically mass-ive] object. But the maths when integrating those equations just blows my mind [and don't get me started on air resistance too!], so I digress! We take the acceleration to be 'g' -9.81ms-2. Now, from maths, we know that the gradient function of a function is its derivative. The rate of change of a function is its derivative, this is important!
Acceleration is 'the rate of change of velocity', so that means 'g' is the rate of change of 'v'. (For purposes of future generalisation, I'll use 'a' instead of 'g'). That also means we can integrate 'a' with respect to time to get 'v'! So v = *Swirly integral S symbol* (a) dt = at + C -- [the constant of integration].
If we were to make a graph of this line, instead of the graph 'y' against 'x', we'll have 'v' against 't'. The constant of integration, C, is where the graph would intercept the 'v' axis. Remember, the 'v' axis is where 't' = 0! what's the velocity when 't' = 0? It's 'u', the initial velocity! [Secondary explosion, watch out; there's shrapnel everywhere! *screams*]
From that, we discovered that the integral of 'a', with respect to time, is 'v', which is equal to 'at + u'. But what's next? I hear you ask. Well, lady/gentleman, let us integrate 'v'! Quickly this time, we know that the velocity is the rate of change of position (NOT DISPLACEMENT, jeez), POSITION! Since we never talk about position, let's give it the symbol 'r'.
*Swirly S* (v) dt = *Another Swirly S* (at + u) dt.
r = (a/2)t2 + ut + C
Now what is C, I hear you ask? Well if you remember the method last time, if we plotted this curve, C would be the initial position! Let's denote that as 'r0'. So, r = (a/2)t2 + ut + r0.
Now this may look different to the equation you see in Physics, but I assure you, this is mathematically sound -- if not, more useful! To prove it, what we use in physics is s = (a/2)t2 + ut, right? and s = final position - initial position, right? Therefore, r - r0 = s = (a/2)t2 + ut, which is what we use! [Wear eye protection people, because your future is suddenly looking bright...]
[...and because of the minds being blown... *boom* poof *boom* *bang**pop*]
In conclusion, these aren't just some random equations vaguely derived from a few assumed values, these are precise mathematical formulae derived from calculus -- the mathematics of change -- representing a projectile being accelerated perfectly downwards on a flat plane, with constant acceleration and no air resistance, in only 1 spatial dimension and 1 time dimension... it doesn't sound too great when I phrase it like that now does it. Oh well. When university comes around, I'm sure I'll be back with my differential equation knowledge to lecture you all once more!
But in all seriousness, if you've discovered anything similar, preferably something more complex, I'd truly be intrigued to know about it! and thanks if you read this far!
TL;DR:
Medical Advice: This thread may cause drowsiness, fatigue and boredom. If you experience any of the symptoms listed above, please leave because you DO NOT BELONG HERE! *shrieks*.
In almost every physics lesson, our class is reassured that this topic 'isn't going to be mathsy', you know, because that's bad right? Physics and Maths are two separate subjects aren't they? They shouldn't be connected in any way whatsoever!? RIGHT?!
WRONG!
I'm pretty 'sad' when it comes to physics -- from others' 'frame of reference' [giggles]. I read a lot about it in my free time and mess about with simple concepts, trying to figure out how they work. In that time, I 'discovered' a few things that we weren't taught that I just think NEEDS to be brought up. These things have greatly increased my understanding of what physics is to me.
In today's lesson, I hope to kick things off with how integration and differentiation are entwined with SUVAT equations!
You know suvat equations right? The boring equations used to calculate the displacement of a particle, its velocity and who knows what other useless things...right? If you seriously think that, then get ready for your eyes to be...opened [explosion].
In all cases in physics AS, we take the acceleration due to gravity to be constant at 'g'. Fundamentally this is wrong, it changes with respect to the distance from the massive [scientifically mass-ive] object. But the maths when integrating those equations just blows my mind [and don't get me started on air resistance too!], so I digress! We take the acceleration to be 'g' -9.81ms-2. Now, from maths, we know that the gradient function of a function is its derivative. The rate of change of a function is its derivative, this is important!
Acceleration is 'the rate of change of velocity', so that means 'g' is the rate of change of 'v'. (For purposes of future generalisation, I'll use 'a' instead of 'g'). That also means we can integrate 'a' with respect to time to get 'v'! So v = *Swirly integral S symbol* (a) dt = at + C -- [the constant of integration].
If we were to make a graph of this line, instead of the graph 'y' against 'x', we'll have 'v' against 't'. The constant of integration, C, is where the graph would intercept the 'v' axis. Remember, the 'v' axis is where 't' = 0! what's the velocity when 't' = 0? It's 'u', the initial velocity! [Secondary explosion, watch out; there's shrapnel everywhere! *screams*]
From that, we discovered that the integral of 'a', with respect to time, is 'v', which is equal to 'at + u'. But what's next? I hear you ask. Well, lady/gentleman, let us integrate 'v'! Quickly this time, we know that the velocity is the rate of change of position (NOT DISPLACEMENT, jeez), POSITION! Since we never talk about position, let's give it the symbol 'r'.
*Swirly S* (v) dt = *Another Swirly S* (at + u) dt.
r = (a/2)t2 + ut + C
Now what is C, I hear you ask? Well if you remember the method last time, if we plotted this curve, C would be the initial position! Let's denote that as 'r0'. So, r = (a/2)t2 + ut + r0.
Now this may look different to the equation you see in Physics, but I assure you, this is mathematically sound -- if not, more useful! To prove it, what we use in physics is s = (a/2)t2 + ut, right? and s = final position - initial position, right? Therefore, r - r0 = s = (a/2)t2 + ut, which is what we use! [Wear eye protection people, because your future is suddenly looking bright...]
[...and because of the minds being blown... *boom* poof *boom* *bang**pop*]
In conclusion, these aren't just some random equations vaguely derived from a few assumed values, these are precise mathematical formulae derived from calculus -- the mathematics of change -- representing a projectile being accelerated perfectly downwards on a flat plane, with constant acceleration and no air resistance, in only 1 spatial dimension and 1 time dimension... it doesn't sound too great when I phrase it like that now does it. Oh well. When university comes around, I'm sure I'll be back with my differential equation knowledge to lecture you all once more!
But in all seriousness, if you've discovered anything similar, preferably something more complex, I'd truly be intrigued to know about it! and thanks if you read this far!
TL;DR:
Spoiler
Medical Advice: This thread may cause drowsiness, fatigue and boredom. If you experience any of the symptoms listed above, please leave because you DO NOT BELONG HERE! *shrieks*.
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Gogo physics students, who can do better on M5? 
Original post by Placeboo123
Gogo physics students, who can do better on M5?
Unfortunately we don't cover M5 where I am, but that sounds like a world of fun :O
Of course there are mathematical aspects of Physics, but to me the most pleasurable parts come from visualising the world or universe at work. You don't need to integrate to derive the suvat equations-all that is needed is simple geometry concerning areas/triangles and graphs (assuming constant acceleration).
Even Einstein's theory of special relativity requires no greater mathematical background than understanding Pythagoras' theorem. The same is true for Snell's law. You can do a surprising amount without resorting to very complex mathematics.
Having said that, the maths is fun too. But A-level physics should be more about intuition than integrals. You can do the integrals in M5 or whatever.
Even Einstein's theory of special relativity requires no greater mathematical background than understanding Pythagoras' theorem. The same is true for Snell's law. You can do a surprising amount without resorting to very complex mathematics.
Having said that, the maths is fun too. But A-level physics should be more about intuition than integrals. You can do the integrals in M5 or whatever.
Original post by Blutooth
Of course there are mathematical aspects of Physics, but to me the most pleasurable parts come from visualising the world or universe at work. You don't need to integrate to derive the suvat equations-all that is needed is simple geometry concerning areas/triangles and graphs (assuming constant acceleration).
Even Einstein's theory of special relativity requires no greater mathematical background than understanding Pythagoras' theorem. The same is true for Snell's law. You can do a surprising amount without resorting to very complex mathematics.
Having said that, the maths is fun too. But A-level physics should be more about intuition than integrals. You can do the integrals in M5 or whatever.
Even Einstein's theory of special relativity requires no greater mathematical background than understanding Pythagoras' theorem. The same is true for Snell's law. You can do a surprising amount without resorting to very complex mathematics.
Having said that, the maths is fun too. But A-level physics should be more about intuition than integrals. You can do the integrals in M5 or whatever.
I know this stuff isn't necessary, and I've already figured out how to derive the lorentz factor, I've just been messing around and been surprised to find that the majority of AS physics students don't know this stuff! They seem to believe that maths has nothing to do with physics when it evidently does!
Also, the furthest my college goes with mechanics is M2 I believe, unfortunately. I've just taken a look at M5 papers and oh my lord, I think I've just found heaven
Physics just seems like a really boring A Level to me, I have no interest in mechanics what so ever, and everything else in the A Level is just covered to mediocre levels of detail. Would love a physics A Level with more Electronics and space physics in it, less mechanics, materials and all that boring stuff. They could also introduce programming in like C/C++ in it, just to a basic understanding level as any modern physics degree has programming in it, and the A Level should try to simulate what it would be like to study physics at university rather than calculating the terminal velocity of a fish before it jumps into water (I mean who actually needs this these days).
The maths in physics is mediocre too, its just basic bidmas stuff, not much in the way of algebra, no use of any calculus (although you could use it for some mechanics questions here and there but it's awkward using it, with the question being easier to work out with equations). I think they should really push the calculus and trigonometry (so differential equations and the like) further, especially at A2. Maths is the best language for physics, and if you don't want to learn it, don't study physics, because you won't get far without maths.
The maths in physics is mediocre too, its just basic bidmas stuff, not much in the way of algebra, no use of any calculus (although you could use it for some mechanics questions here and there but it's awkward using it, with the question being easier to work out with equations). I think they should really push the calculus and trigonometry (so differential equations and the like) further, especially at A2. Maths is the best language for physics, and if you don't want to learn it, don't study physics, because you won't get far without maths.
The clue is in the tilte: G.C.E. A-level
i.e. General Certificate in Education.
A-level courses have to be all things to all people. Most people who study physics at A-level do not want or need to study it at degree or above, but need to understand basic concepts to take forward into their chosen degrees.
It is far more important to understand the concepts at this level than to get bogged down in complex maths as well.
A-levels are structured to try and not overlap between subjects to avoid duplication and wasted time going over the same stuff which many people will bitch about if that happened. Some of that is unavoidable though. Damned if you do, damned if you don't.
The biggest problem is people wanting to run before they can walk. Basic laboratory skills are woefully lacking because not enough time is spent in the lab.
Electricity and magnetism is a huge problem for many people simply because it's hard to visualise what's going on and many simply plug and chug equations without having any real understanding of what the equations mean or how to interpret results. Interestingly, I find arts people have an intuitive grasp in this topic (if they also take physics) because they have a spatial awareness lacking in many STEM students.
Any programming language can be learned in ones own time from any decent book and there are lot's out there. teach yourself C++ in 21 days by Aitken and Jones for instance. However, to programme well, one needs structured techniques like Object Oriented, Yourdon, Ward-Mellor etc. which can take up a whole module at degree level in their own right.
People also need to realise that there is far less hand holding at degree level and much of the learning is done off the undergraduates own backs, supplemented by lectures and seminars. So if one wants to know what it's like as an undergrad in physics, there is nothing stopping you. "Pick up thy sticks and walk!"
The first year of many degree courses tend to concentrate on bringing a diverse background of students up to a common level of understanding which covers much of the A-level syllabus in more depth and adds a few more topics. It's not until one gets into the third year that the meaty stuff is covered.
"The paradox is only a conflict between reality and your feeling of what reality ought to be" -Richard Feynman
i.e. General Certificate in Education.
A-level courses have to be all things to all people. Most people who study physics at A-level do not want or need to study it at degree or above, but need to understand basic concepts to take forward into their chosen degrees.
It is far more important to understand the concepts at this level than to get bogged down in complex maths as well.
A-levels are structured to try and not overlap between subjects to avoid duplication and wasted time going over the same stuff which many people will bitch about if that happened. Some of that is unavoidable though. Damned if you do, damned if you don't.
The biggest problem is people wanting to run before they can walk. Basic laboratory skills are woefully lacking because not enough time is spent in the lab.
Electricity and magnetism is a huge problem for many people simply because it's hard to visualise what's going on and many simply plug and chug equations without having any real understanding of what the equations mean or how to interpret results. Interestingly, I find arts people have an intuitive grasp in this topic (if they also take physics) because they have a spatial awareness lacking in many STEM students.
Any programming language can be learned in ones own time from any decent book and there are lot's out there. teach yourself C++ in 21 days by Aitken and Jones for instance. However, to programme well, one needs structured techniques like Object Oriented, Yourdon, Ward-Mellor etc. which can take up a whole module at degree level in their own right.
People also need to realise that there is far less hand holding at degree level and much of the learning is done off the undergraduates own backs, supplemented by lectures and seminars. So if one wants to know what it's like as an undergrad in physics, there is nothing stopping you. "Pick up thy sticks and walk!"
The first year of many degree courses tend to concentrate on bringing a diverse background of students up to a common level of understanding which covers much of the A-level syllabus in more depth and adds a few more topics. It's not until one gets into the third year that the meaty stuff is covered.
"The paradox is only a conflict between reality and your feeling of what reality ought to be" -Richard Feynman
(edited 9 years ago)
Original post by Callum Scott
In all cases in physics AS, we take the acceleration due to gravity to be constant at 'g'. Fundamentally this is wrong, it changes with respect to the distance from the massive [scientifically mass-ive] object.stance...]
In all cases in physics AS, we take the acceleration due to gravity to be constant at 'g'. Fundamentally this is wrong, it changes with respect to the distance from the massive [scientifically mass-ive] object.stance...]
Yes and it also depends on your geographical position as the Earth is not a perfect sphere and not of uniform density. It also varies as a result of the Earth's spin, having a greater measured value at the poles compared to the equator.
Here's a question for you.
Assuming the value of g at the surface at your location happens to be 9.81m/s/s
a) by how much does it vary between the top and bottom of a tower, 20m high, when you drop an object to the ground. State your assumptions.
b) From your answer to a) argue logically why we do not normally consider this difference when answering questions about the final velocity when it reaches the ground.
Original post by Stonebridge
Yes and it also depends on your geographical position as the Earth is not a perfect sphere and not of uniform density. It also varies as a result of the Earth's spin, having a greater measured value at the poles compared to the equator.
Here's a question for you.
Assuming the value of g at the surface at your location happens to be 9.81m/s/s
a) by how much does it vary between the top and bottom of a tower, 20m high, when you drop an object to the ground. State your assumptions.
b) From your answer to a) argue logically why we do not normally consider this difference when answering questions about the final velocity when it reaches the ground.
Here's a question for you.
Assuming the value of g at the surface at your location happens to be 9.81m/s/s
a) by how much does it vary between the top and bottom of a tower, 20m high, when you drop an object to the ground. State your assumptions.
b) From your answer to a) argue logically why we do not normally consider this difference when answering questions about the final velocity when it reaches the ground.
I fully understand why we don't cover differential calculus at AS level, I know that it's unreasonable to consider the Earth in its true form as it'd require a supercomputer to work that stuff out! I never even proposed that we learn how to factor in air resistance! All I said is that there are many other factors that we ignore, and proceeded to enlighten AS physics students that calculus and physics are so obviously related.
I was proposing that for people to take both AS maths and physics, you can interweave the two subjects to get a better fundamental understanding of its concepts.
From the responses, I feel the need to reiterate. In no way am I proposing they change the education system, I'm trying to allow other physics students like me, at AS level, to see that they (your AS maths and physics classes) are related.
When I say stuff like velocity is the rate of change of position, not displacement...I'm just pointing out that from the derivation above, something that we're not taught at AS pops out, something I find to more intuitiv and easier to handle.
Original post by uberteknik
try and not overlap between subjects to avoid duplication and wasted time going over the same stuff which many people will bitch about if that happened. Some of that is unavoidable though. Damned if you do, damned if you don't.
--------------
People also need to realise that there is far less hand holding at degree level and much of the learning is done off the undergraduates own backs, supplemented by lectures and seminars. So if one wants to know what it's like as an undergrad in physics, there is nothing stopping you. "Pick up thy sticks and walk!"
try and not overlap between subjects to avoid duplication and wasted time going over the same stuff which many people will bitch about if that happened. Some of that is unavoidable though. Damned if you do, damned if you don't.
--------------
People also need to realise that there is far less hand holding at degree level and much of the learning is done off the undergraduates own backs, supplemented by lectures and seminars. So if one wants to know what it's like as an undergrad in physics, there is nothing stopping you. "Pick up thy sticks and walk!"
I'm not saying they should change the syllabus to incorporate more maths. I'm not proposing that we teach all physics students calculus, despite the chance they might have already done it.
What I said was, if you took both physics and maths, you can incorporate your maths knowledge into physics too! And then my example: basic calculus can be used to derive the suvat equations and to try and improve one's understanding and concept of them.
Other people can then extrapolate and beg the question, what If acceleration wasn't constant? What if changed? What if there were 2 spatial dimensions, or 3, or 4? Well we just integrate a linearly increasing acceleration or whatever. I want people who already have an interest in maths to 'bridge the gap' and figure this stuff out because at the moment, we're taught the suvats like they're some fundamental law explaining any given object in any situation (though I guess air resistance is shoved down our throats a lot), and that's all that maths students think they are!
They may go deeper in explanation in A level, but I don't know, and what I care about is opening people's eyes (AS physics and maths students') to see that the 2 subjects they chose are related and you can get a better understanding of physics because of it.
Original post by Thahleel
Physics just seems like a really boring A Level to me, I have no interest in mechanics what so ever, and everything else in the A Level is just covered to mediocre levels of detail. Would love a physics A Level with more Electronics and space physics in it, less mechanics...
Personally, I agree with almost everything you've said lol. Computational physics would be awesome, but that would take up most of the time in the course and we'd have no time to understand how it worked.
Again, personally, I'd love to be taught orbital mechanics, non-circular motion, more applications for calculus, special and general relativity, quantum mechanics; all from the get-go. But I know for a fact this is completely unreasonable as the majority of the students will have no idea what this stuff is and will have no use for it whatsoever.
So in my free time, I fiddle with the stuff myself! I much prefer the knowledge of how the suvats work than the reassurance that they just do. I much prefer a beautiful derivation that takes 3 seconds to one that requires minutes of contemplation. I'd like to be taught that acceleration is the second derivative of position than that it's the rate of change of the rate of change of displacement. But I think that'd be too much for other non-maths students and alas, we are forced to find it out for ourselves - which is where I hope to help others!
(edited 9 years ago)
Original post by Callum Scott
I was proposing that for people to take both AS maths and physics, you can interweave the two subjects to get a better fundamental understanding of its concepts.
From the responses, I feel the need to reiterate. In no way am I proposing they change the education system, I'm trying to allow other physics students like me, at AS level, to see that they (your AS maths and physics classes) are related.
When I say stuff like velocity is the rate of change of position, not displacement...I'm just pointing out that from the derivation above, something that we're not taught at AS pops out, something I find to more intuitiv and easier to handle.
I was proposing that for people to take both AS maths and physics, you can interweave the two subjects to get a better fundamental understanding of its concepts.
From the responses, I feel the need to reiterate. In no way am I proposing they change the education system, I'm trying to allow other physics students like me, at AS level, to see that they (your AS maths and physics classes) are related.
When I say stuff like velocity is the rate of change of position, not displacement...I'm just pointing out that from the derivation above, something that we're not taught at AS pops out, something I find to more intuitiv and easier to handle.
Why doesn't history teach about great scientific discoveries and the impact it's had on humanity?
Why doesn't English teach how to write excellent science reports in an engaging and interesting style?
Why doesn't music cover the mathematical theory of waves and harmonics, Binomial expansions, Fourier analysis, etc?
Why doesn't Modern Foreign Languages teach the Greek Alphabet used throughout maths and the sciences?
A: because a line has to be drawn and compromises made. They may not all be optimised compromises, but IMHO they do a reasonable job of covering most bases in an efficient way.
The types of links you suggested, were indeed incorporated into the syllabus (O and A-levels pre 1985). The difficulty is that not all students have the same ability and whilst some will find it relatively easy, others struggled because of the extra layer of mathematics needed. That meant slower starters automatically penalised because a higher level of maths is needed to even begin the physics course - which put many off, even though they were easily capable but needed a bit more time.
Ironically, putting people off is exactly the opposite effect of trying to engage people with science by incorporating more maths.
The students that have a desire to study science at a higher level, will inevitably be required to acquire both maths and physics before entering university.
It's not perfect, but then to paraphrase an Abraham Lincoln quotation:
"One can please some of the people all of time and all of the people some of the time, but not all of the people all of the time".
(edited 9 years ago)
Original post by uberteknik
The types of links you suggested, were indeed incorporated into the syllabus (O and A-levels pre 1985). The difficulty is that not all students have the same ability and whilst some will find it relatively easy, others struggled because of the extra layer of mathematics needed. That meant slower starters automatically penalised because a higher level of maths is needed to even begin the physics course - which put many off, even though they were easily capable but needed a bit more time.
I don't want to re-implement that system, I don't want all students to know about this stuff, I love physics and maths and I spend time bridging the gap to make more sense of both of the subjects. All I'm saying is that if there are people on here that, like me, are taking AS maths and physics and, like me, enjoy them thoroughly, here's a neat little tunnel that links them together that we weren't taught, can anyone open my eyes to any more? This thread wasn't mainly a question, it was simply a post for the people out there like me, because I found it pretty cool and equally disappointing that no one else in my physics class or any others at AS level, noticed.
I just wanted to know what else, from the subjects we've covered at AS physics, like my example, isn't fully mathematically explained that I'd enjoy reading more into.
An interesting question inspired by your post is the following.
Q: Say the Earth is a perfect sphere but, a non-uniform density. Now, constrained to the surface of the Earth, does the gravitational field vary with respect to your location?
Q: Say the Earth is a perfect sphere but, a non-uniform density. Now, constrained to the surface of the Earth, does the gravitational field vary with respect to your location?
Original post by Stonebridge
Yes and it also depends on your geographical position as the Earth is not a perfect sphere and not of uniform density. It also varies as a result of the Earth's spin, having a greater measured value at the poles compared to the equator.
Here's a question for you.
Assuming the value of g at the surface at your location happens to be 9.81m/s/s
a) by how much does it vary between the top and bottom of a tower, 20m high, when you drop an object to the ground. State your assumptions.
b) From your answer to a) argue logically why we do not normally consider this difference when answering questions about the final velocity when it reaches the ground.
Here's a question for you.
Assuming the value of g at the surface at your location happens to be 9.81m/s/s
a) by how much does it vary between the top and bottom of a tower, 20m high, when you drop an object to the ground. State your assumptions.
b) From your answer to a) argue logically why we do not normally consider this difference when answering questions about the final velocity when it reaches the ground.
Original post by WishingChaff
An interesting question inspired by your post is the following.
Q: Say the Earth is a perfect sphere but, a non-uniform density. Now, constrained to the surface of the Earth, does the gravitational field vary with respect to your location?
Q: Say the Earth is a perfect sphere but, a non-uniform density. Now, constrained to the surface of the Earth, does the gravitational field vary with respect to your location?
It will depend on how the density variations vary with location within the Earth.
What makes you think that?
Original post by Stonebridge
It will depend on how the density variations vary with location within the Earth.
Original post by Callum Scott
*This thread is purely for people who chose both Maths and Physics*
In almost every physics lesson, our class is reassured that this topic 'isn't going to be mathsy', you know, because that's bad right? Physics and Maths are two separate subjects aren't they? They shouldn't be connected in any way whatsoever!? RIGHT?!
WRONG!
I'm pretty 'sad' when it comes to physics -- from others' 'frame of reference' [giggles]. I read a lot about it in my free time and mess about with simple concepts, trying to figure out how they work. In that time, I 'discovered' a few things that we weren't taught that I just think NEEDS to be brought up. These things have greatly increased my understanding of what physics is to me.
In today's lesson, I hope to kick things off with how integration and differentiation are entwined with SUVAT equations!
You know suvat equations right? The boring equations used to calculate the displacement of a particle, its velocity and who knows what other useless things...right? If you seriously think that, then get ready for your eyes to be...opened [explosion].
In all cases in physics AS, we take the acceleration due to gravity to be constant at 'g'. Fundamentally this is wrong, it changes with respect to the distance from the massive [scientifically mass-ive] object. But the maths when integrating those equations just blows my mind [and don't get me started on air resistance too!], so I digress! We take the acceleration to be 'g' -9.81ms-2. Now, from maths, we know that the gradient function of a function is its derivative. The rate of change of a function is its derivative, this is important!
Acceleration is 'the rate of change of velocity', so that means 'g' is the rate of change of 'v'. (For purposes of future generalisation, I'll use 'a' instead of 'g'). That also means we can integrate 'a' with respect to time to get 'v'! So v = *Swirly integral S symbol* (a) dt = at + C -- [the constant of integration].
If we were to make a graph of this line, instead of the graph 'y' against 'x', we'll have 'v' against 't'. The constant of integration, C, is where the graph would intercept the 'v' axis. Remember, the 'v' axis is where 't' = 0! what's the velocity when 't' = 0? It's 'u', the initial velocity! [Secondary explosion, watch out; there's shrapnel everywhere! *screams*]
From that, we discovered that the integral of 'a', with respect to time, is 'v', which is equal to 'at + u'. But what's next? I hear you ask. Well, lady/gentleman, let us integrate 'v'! Quickly this time, we know that the velocity is the rate of change of position (NOT DISPLACEMENT, jeez), POSITION! Since we never talk about position, let's give it the symbol 'r'.
*Swirly S* (v) dt = *Another Swirly S* (at + u) dt.
r = (a/2)t2 + ut + C
Now what is C, I hear you ask? Well if you remember the method last time, if we plotted this curve, C would be the initial position! Let's denote that as 'r0'. So, r = (a/2)t2 + ut + r0.
In almost every physics lesson, our class is reassured that this topic 'isn't going to be mathsy', you know, because that's bad right? Physics and Maths are two separate subjects aren't they? They shouldn't be connected in any way whatsoever!? RIGHT?!
WRONG!
I'm pretty 'sad' when it comes to physics -- from others' 'frame of reference' [giggles]. I read a lot about it in my free time and mess about with simple concepts, trying to figure out how they work. In that time, I 'discovered' a few things that we weren't taught that I just think NEEDS to be brought up. These things have greatly increased my understanding of what physics is to me.
In today's lesson, I hope to kick things off with how integration and differentiation are entwined with SUVAT equations!
You know suvat equations right? The boring equations used to calculate the displacement of a particle, its velocity and who knows what other useless things...right? If you seriously think that, then get ready for your eyes to be...opened [explosion].
In all cases in physics AS, we take the acceleration due to gravity to be constant at 'g'. Fundamentally this is wrong, it changes with respect to the distance from the massive [scientifically mass-ive] object. But the maths when integrating those equations just blows my mind [and don't get me started on air resistance too!], so I digress! We take the acceleration to be 'g' -9.81ms-2. Now, from maths, we know that the gradient function of a function is its derivative. The rate of change of a function is its derivative, this is important!
Acceleration is 'the rate of change of velocity', so that means 'g' is the rate of change of 'v'. (For purposes of future generalisation, I'll use 'a' instead of 'g'). That also means we can integrate 'a' with respect to time to get 'v'! So v = *Swirly integral S symbol* (a) dt = at + C -- [the constant of integration].
If we were to make a graph of this line, instead of the graph 'y' against 'x', we'll have 'v' against 't'. The constant of integration, C, is where the graph would intercept the 'v' axis. Remember, the 'v' axis is where 't' = 0! what's the velocity when 't' = 0? It's 'u', the initial velocity! [Secondary explosion, watch out; there's shrapnel everywhere! *screams*]
From that, we discovered that the integral of 'a', with respect to time, is 'v', which is equal to 'at + u'. But what's next? I hear you ask. Well, lady/gentleman, let us integrate 'v'! Quickly this time, we know that the velocity is the rate of change of position (NOT DISPLACEMENT, jeez), POSITION! Since we never talk about position, let's give it the symbol 'r'.
*Swirly S* (v) dt = *Another Swirly S* (at + u) dt.
r = (a/2)t2 + ut + C
Now what is C, I hear you ask? Well if you remember the method last time, if we plotted this curve, C would be the initial position! Let's denote that as 'r0'. So, r = (a/2)t2 + ut + r0.
R-C is displacement is s so that is the physics equation
Original post by WishingChaff
An interesting question inspired by your post is the following.
Q: Say the Earth is a perfect sphere but, a non-uniform density. Now, constrained to the surface of the Earth, does the gravitational field vary with respect to your location?
Q: Say the Earth is a perfect sphere but, a non-uniform density. Now, constrained to the surface of the Earth, does the gravitational field vary with respect to your location?
You can model as acting from COM so if the COM was not in the centre then you would get closer when you moved round so yes if COM is not at centre of sphere
Original post by WishingChaff
An interesting question inspired by your post is the following.
Q: Say the Earth is a perfect sphere but, a non-uniform density. Now, constrained to the surface of the Earth, does the gravitational field vary with respect to your location?
Q: Say the Earth is a perfect sphere but, a non-uniform density. Now, constrained to the surface of the Earth, does the gravitational field vary with respect to your location?
Obviously it does. Suppose in case A) almost all of the mass were concentrated at a point far from where you were. Suppose in B) almost all of the mass was were concentrated right below where you were standing. Clearly, there would a difference in the gravitational pull to the centre of the earth in cases A and B.
Original post by Random1357
You can model as acting from COM so if the COM was not in the centre then you would get closer when you moved round so yes if COM is not at centre of sphere
I think what you say is not right. Image a sphere fitted inside a tetrahedron, where the sphere touches the edges of the tetrahedron. Call these points intersection points. Suppose the sphere is such that its mass is very densely concentrated at these intersections. g will be much stronger near these intersections, than it will be at other points around the sphere and yet its centre of mass will be at the centre of the sphere.
(edited 9 years ago)
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