Physics is nothing without math.
Physics is all about derivatives, integrals, cross and dot products; it is the application of math. Here are some ways that understanding math will help with physics:
Derivatives are all about the rate of change of something. When you are taking the derivative of something, you are generally trying to see how much it changes with respect to time. Identify exactly what is changing, and you will be able to simplify the equations. Once you know the fundamental concept and memorize the main formula that goes with it, it is much easier to derive other equations in the quantities that are given in a situation. Most equations are variations of another equation, or are derived from something simpler. Once you understand how all the concepts link together, you don't need to memorize all the other variations. It will come to you and make sense!
Integrals are basically the reverse of derivatives. If you know the rate of change of something and want to find out what that value is at an instant of time (like taking a snapshot), integrating the formula with respect to whatever you want to "pause" (this could be time or space) it at will give you that instantaneous quantity. This comes in handy when you want to know what is happening at a particular instance.
Cross products tell you what direction things are acting in. You get a vector quantity. This is for 3D application, usually the two things you multiply together are on the same plane and their cross product gives you the direction of the third plane.
Dot products are a scalar quantity meaning there is no direction for the product of whatever you multiplied together.
Some Extra Tips:
1. Try and link concepts together. Are there similar formulas for different things? Patterns?
2. General quantities are Force, Energy, Work. All concepts carry over from one type of physics to another. Newton's three laws are consistent no matter what kind of forces we are dealing with. Are there multiple ways to find the quantity in question? Remember: the number of unknowns you are looking for requires that same number of equations. So if you have more things you need to find than equations you have to find it, there might be something you're missing! Sometimes the examiner will give you extra quantities that you will not need to use, so be wary of these.
3. Be consistent. It is much easier to memorize formulas that consist of all multiplication. Division is just a subset of multiplication anyway! For example, it is much easier to remember voltage as the product of current and resistance ( V = IR), rather than remembering that current is the quotient of voltage over resistance. Sometimes it can get confusing to remember which quantity goes on top. Although multiplication is much easier to remember, there are always exceptions! For example, things that are derivatives of something else are much easier to remember by division, as that's what derivatives are! It is easier to remember that velocity is the rate of change of distance (v = d/t) than to remember that distance is the product of velocity and time. Understanding what the mathematical terms mean is crucial to understanding the physics of how things work, leading to a mindset that understands physics.
4. If you understand the concept, the formulas will be easier to remember. Are you memorizing the formulas, or do you actually understand them? Understanding things last much longer than just memorizing something. I have a terrible memory, so anything I "cram" into my brain the night before an exam only stays until that exam is over. However, that is risky as I might blank out during the exam and forget everything. Understanding the concept is much better because you will be able to know what the formula should generally look like, and once you have it correct, it will look familiar to you.
5. Physics is all around. Once you observe something, it stays with you. Thinking about why and how something works always contributes. Physics can always be verified with experiments. What happens if you change one thing?? How much is something affected? Does this make sense? Thinking about these sorts of things will help you know if your answers are correct.
6. Units. It is always helpful to know what the unit of measurement is. This is especially helpful when you forget a formula. You are given some quantities, you are looking for another quantity. They are somehow related through an equation. Sometimes, you can figure out what this equation is by just looking at the units.
For example, if I am looking for velocity (m/s) and am given the position (m) and time (s), looking at the units for velocity automatically gives me the formula to find it. Another thing about units is to make sure that all the units are consistent. This is usually something that a lot of people can overlook and make a big difference in the answer. Make sure that you have everything in the same units. It is helpful to multiply, divide and cross out your units to make sure that you have the right formula.
I am writing these things because I have a physics exam on Thursday. I have just finished reviewing everything from the whole course and wrote out a formula sheet that I have to remember. These are just some things that I noticed while I was making this sheet and trying to make my life easier during the exam. I have a terrible memory. I make a lot of silly mistakes. I hope that these tips and relationships between math and physics will help you in discovering that joy in understanding something that seemed like gibberish before. My only hope is that I will actually do well on my exam so that I can say this and not look stupid. Anyway, it is always much easier to know what to do than to actually do it. So if I don't do well, it is because I didn't follow my own advice.
Today I read Hebrews 5. It warns against falling away from God because of lack of obedience to Him.
Verses 11 - 14 say this, "We have much to say about this, but it is hard to make it clear to you because you no longer try to understand. In fact, though by this time you ought to be teachers, you need someone to teach you the elementary truths of God’s word all over again. You need milk, not solid food! Anyone who lives on milk, being still an infant, is not acquainted with the teaching about righteousness. But solid food is for the mature, who by constant use have trained themselves to distinguish good from evil."
Just as in physics, you need to understand the basics, the elementary truths in order to understand harder concepts. Likewise, you need to read the Bible to know what is good and evil in God's eyes. Once you know the math behind it, you can apply it in the form of physics. Once you know what is good and evil, you can do it. You have to train yourself to be good at something. You have to do a lot of practice problems before you can jump ahead. You have to do the right thing when it is obvious in order for you to know what is right in more difficult situations.
Physics is all about derivatives, integrals, cross and dot products; it is the application of math. Here are some ways that understanding math will help with physics:
Derivatives are all about the rate of change of something. When you are taking the derivative of something, you are generally trying to see how much it changes with respect to time. Identify exactly what is changing, and you will be able to simplify the equations. Once you know the fundamental concept and memorize the main formula that goes with it, it is much easier to derive other equations in the quantities that are given in a situation. Most equations are variations of another equation, or are derived from something simpler. Once you understand how all the concepts link together, you don't need to memorize all the other variations. It will come to you and make sense!
Integrals are basically the reverse of derivatives. If you know the rate of change of something and want to find out what that value is at an instant of time (like taking a snapshot), integrating the formula with respect to whatever you want to "pause" (this could be time or space) it at will give you that instantaneous quantity. This comes in handy when you want to know what is happening at a particular instance.
Cross products tell you what direction things are acting in. You get a vector quantity. This is for 3D application, usually the two things you multiply together are on the same plane and their cross product gives you the direction of the third plane.
Dot products are a scalar quantity meaning there is no direction for the product of whatever you multiplied together.
Some Extra Tips:
1. Try and link concepts together. Are there similar formulas for different things? Patterns?
2. General quantities are Force, Energy, Work. All concepts carry over from one type of physics to another. Newton's three laws are consistent no matter what kind of forces we are dealing with. Are there multiple ways to find the quantity in question? Remember: the number of unknowns you are looking for requires that same number of equations. So if you have more things you need to find than equations you have to find it, there might be something you're missing! Sometimes the examiner will give you extra quantities that you will not need to use, so be wary of these.
3. Be consistent. It is much easier to memorize formulas that consist of all multiplication. Division is just a subset of multiplication anyway! For example, it is much easier to remember voltage as the product of current and resistance ( V = IR), rather than remembering that current is the quotient of voltage over resistance. Sometimes it can get confusing to remember which quantity goes on top. Although multiplication is much easier to remember, there are always exceptions! For example, things that are derivatives of something else are much easier to remember by division, as that's what derivatives are! It is easier to remember that velocity is the rate of change of distance (v = d/t) than to remember that distance is the product of velocity and time. Understanding what the mathematical terms mean is crucial to understanding the physics of how things work, leading to a mindset that understands physics.
4. If you understand the concept, the formulas will be easier to remember. Are you memorizing the formulas, or do you actually understand them? Understanding things last much longer than just memorizing something. I have a terrible memory, so anything I "cram" into my brain the night before an exam only stays until that exam is over. However, that is risky as I might blank out during the exam and forget everything. Understanding the concept is much better because you will be able to know what the formula should generally look like, and once you have it correct, it will look familiar to you.
5. Physics is all around. Once you observe something, it stays with you. Thinking about why and how something works always contributes. Physics can always be verified with experiments. What happens if you change one thing?? How much is something affected? Does this make sense? Thinking about these sorts of things will help you know if your answers are correct.
6. Units. It is always helpful to know what the unit of measurement is. This is especially helpful when you forget a formula. You are given some quantities, you are looking for another quantity. They are somehow related through an equation. Sometimes, you can figure out what this equation is by just looking at the units.
For example, if I am looking for velocity (m/s) and am given the position (m) and time (s), looking at the units for velocity automatically gives me the formula to find it. Another thing about units is to make sure that all the units are consistent. This is usually something that a lot of people can overlook and make a big difference in the answer. Make sure that you have everything in the same units. It is helpful to multiply, divide and cross out your units to make sure that you have the right formula.
I am writing these things because I have a physics exam on Thursday. I have just finished reviewing everything from the whole course and wrote out a formula sheet that I have to remember. These are just some things that I noticed while I was making this sheet and trying to make my life easier during the exam. I have a terrible memory. I make a lot of silly mistakes. I hope that these tips and relationships between math and physics will help you in discovering that joy in understanding something that seemed like gibberish before. My only hope is that I will actually do well on my exam so that I can say this and not look stupid. Anyway, it is always much easier to know what to do than to actually do it. So if I don't do well, it is because I didn't follow my own advice.
Today I read Hebrews 5. It warns against falling away from God because of lack of obedience to Him.
Verses 11 - 14 say this, "We have much to say about this, but it is hard to make it clear to you because you no longer try to understand. In fact, though by this time you ought to be teachers, you need someone to teach you the elementary truths of God’s word all over again. You need milk, not solid food! Anyone who lives on milk, being still an infant, is not acquainted with the teaching about righteousness. But solid food is for the mature, who by constant use have trained themselves to distinguish good from evil."
Just as in physics, you need to understand the basics, the elementary truths in order to understand harder concepts. Likewise, you need to read the Bible to know what is good and evil in God's eyes. Once you know the math behind it, you can apply it in the form of physics. Once you know what is good and evil, you can do it. You have to train yourself to be good at something. You have to do a lot of practice problems before you can jump ahead. You have to do the right thing when it is obvious in order for you to know what is right in more difficult situations.
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