I find that students are not comfortable with the idea of physics because it is due to two things. Either they do not have enough mathematical background or they need more physical intuition. Really most students are unable to do simple algebra problems and expect to do well in physics or are unfamiliar with some trigonometry which would make the concepts difficult to understand. This post is really to share what math is needed to do high school physics and how they come in.
The first topic I want to discuss is the simple idea of algebra. This is not anything fancy just given an equation, we want to solve for some other variable. This comes when we take a look at the basic equations of kinematics. Consider just the basic equation of kinematics which is the change in position is equation to the initial velocity times time added to half the acceleration multiplied by the square of the time taken. This requires a lot of algebra since most of the time we are not given one of the variables velocity, position, time, nor acceleration. Being able to rearrange equations so they equal something else is valuable since by understanding algebra one would be able to find different unknowns while only knowing one equation.
Trigonometry is helpful in that it really links two things together. Most of the time, the equations of motion studied in high school are only in one dimension and thus when there are more than one dimension it becomes very difficult to solve the equations of motion. However this can be fixed by breaking things into two one dimension problems then solving and relating the dimensions with the ideas of geometry. This idea is used to kind of change our point of view on a questions and realizing that no matter how we are looking at the problem, the results are the same. We call this invariant by coordinate transformations. Thus by using trigonometry, we can do these invariant transformations and see that the equations of motion in high school physics, are the same.
Now after the study of high school physics we come with more difficult problems, where we want to generalize everything that we have learned into a few basic equations and be able to re derive everything just by looking at special cases. This we need the ideas of matrices because we are not just breaking the problem into two equations but sometimes we need to break the equations into more forms where solving the equations of motion is just using what we have learned about matrices. With that in mind we see that all of general physics can be generalized just with cases of one value being different. For example, when we write the equations of motion in a general form, taking account to energy conservations, we find that most periodic motion is dependent on something called the eigenvalues of the matrix of the system. With just changes in these values we could either have motion that goes to infinity like throwing a ball into space to motion that follows a nice path, to periodic motion.
Now there are many more things to examine while what I talked about today is just the tip of the iceberg. Further down in physics we would need larger more powerful mathematics such as group theory, for particle physics, tensor analysis, for general relativity, calculus of variation, for whatever suites your taste, and even higher topics such as c star algebra. So I hope this has been insightful where to understand physics we need to understand a lot more math.
This article was written for you by Brandon, one of the tutors with SchoolTutoring Academy.