If you can solve these five equations, your algebra skills are strong enough that you should feel confident entering your first physics class!
But what if you don't get these correct?! Don't worry! Most students entering their first year of physics will get the majority of these incorrect at first. This is why you need to brush up on your algebra skills prior to entering physics class. A little review of the basics can go a long way!
Let's go over each of these examples and locate some of the common spots where students get off track.
Before we start, recall the basic order of operations (parenthesis, exponents, multiplication/division, addition/subtraction). When solving for a variable, we do the order of operations in reverse. Imagine keeping the variable that you want to solve for where it is, and slowly moving the other variables to the other side. If they are attached to the variable you are solving for through addition, use subtraction to move it to the other side. (If this doesn't sound familiar to you, visit Khan Academy - Algebra Basics to brush up on your fundamental algebra skills.)
You may take a different path than I did when solving these equations. That is ok! As long as you are following the order of operations, we will arrive at the same ending point.
There is typically more than one path to get to the final answer in physics... and life. I encourage students to not only try different paths, but to discuss how multiple paths are able to get us to the same place. This is where we can start to embrace our differences in mental processing and identify methods that "make more sense" to us. Be open-minded to various paths of problem solving. It may lead you on a whole new adventure!
Let's look at the first equation. It is pretty straight forward. Expect a lot of algebra at this difficulty level when starting. Be sure to use parenthesis when you need to clarify the order of operations. It makes it easier for someone to read and it helps to make sure you don't make a silly error when you substitute in the numbers and do the calculation.
For the next equation, there are two things to watch out for. First, the variable that you want to solve for is in the denominator. You need to get it out of the denominator to solve for it! Second, the variable you are solving for is squared, when you take the square root, your answer mathematically could be positive or negative that number. Sometimes the negative value is nonsensical, but not always!
Note: The "G" could be placed in front of the fraction or it could be in the numerator of the fraction. Mathematically it is processed the same way.
Because this next equation has a t and t squared (and we are tasked with solving for t), you will need to use the quadratic equation to solve it. Note: Sometimes we luck out and another variable in the equation is zero. When that is the case, I simplify the equation prior to doing the algebra. Then, I may not have to use the quadratic equation to solve for t! I may only be left with a t OR a t squared.
Solving for an angle is not complex. You just need to remember that in order to remove the sine from theta, you will need to take the inverse sine of both sides. Just like multiplication is undone with division, sine is undone by inverse sine. When it comes to putting numbers into your calculator, be sure you are in the correct mode, degree or radian, when calculating!
OK! This last one may not show up in your first year of physics (but it has in a course that I have taught). You need to remember how to use the natural logarithm rules.
You may be thinking "Do I really need to rearrange the equation BEFORE I plug in the numbers?" The answer is YES! Rearrange the equation with the variables first! Plugging in numbers is the last step.
Why? Because you are determining the relationship between the variables regardless of specific numerical values. What if you need to redo the problem with different numbers? Also, you are less likely to make a calculator error if you leave all of the number crunching to the last step. Finally, it is quicker and more efficient to use short variables instead of numbers with multiple digits when you are continually rewriting the equation. Why not use "t" instead of "32.14 s"? Trust me, it may be uncomfortable at first manipulating variables instead of numbers, but it will pay off big time in the end!