Motion can be described in three main ways: algebraically, graphically, and narratively.
Motion can be described in three main ways: algebraically, graphically, and narratively. In AP Physics 1, you will be expected to do all three. Episode 2 takes you ice skating to explain graphical relationships - linear (1:40), quadratic (3:06), inverse (4:40) and inverse-square function.(5:40) Once you read the problem, I strongly suggest drawing a picture that diagrams what is known about the object or objects involved (6:22).
The Question of the Day asks (7:57) Newton’s 2nd Law is often written F = m*a. If the force on an object remains constant, and the mass is doubled, what happens to the acceleration?
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Hi and welcome to the APsolute recAP: Physics 1 Edition. Today’s episode will introduce mathematic parent functions and explain problem-solving techniques to help ensure your success.
Lets Zoom out:
Hopefully, it is not news to you that physics utilizes the language of mathematics to help describe our world. In math it is pretty standard to use X’s and Y’s to represent variables within a function. Of course, math problems get cute and throw in “f of x” or “g of f of x”, but usually… X and Y. In physics, you get a solid understanding of what a variable is and the plethora of physical quantities that can be represented with them. As a result of all of the letters flying around in our equations, it is helpful to find ways to organize your work so you don’t lose track of what is and is not useful. We don’t want to place too much emphasis on creating a stepwise process that is rigid and doesn’t allow deviation, instead we want to get in a space where physics problem solving becomes fun and feels like we are solving a riddle or puzzle. Motion can be described in three main ways: algebraically, graphically, and narratively. In AP Physics 1, you will be expected to do all three.
Lets Zoom in:
While hanging out with friends at a local ice skating rink, you push your friend and send them gliding across the ice at a constant speed. Every second that passes, your friend glides a meter farther away. Being that the surface is essentially frictionless, your friend continues to coast a good while. You could probably predict pretty accurately where your friend will be after 5 seconds… they would be 5 meters away. Although I provided you with a descriptive narrative of the motion, your prediction is likely based on your graphical understanding of linear relationships.
Graphs that have a consistent slope are known as linear relationships, and are encountered often in AP Physics 1. We even take more complex relationships, and turn them into linear relationships because straight lines are so much easier to understand than some of the others. Most of us have a wonderful math teacher to thank for also being able to use an equation to describe a linear relationship. Although there are a few ways to write the equation of a line, my personal favorite is slope-intercept form. Good Ole’ y = mx + b! In the ice skating example “y” would be the ending location, “m” would be the speed of the skater, “x” would be the time that has elapsed from start to finish, and “b” would be the initial location or 0 m. Admittedly, we wouldn’t use these variables since they don’t make much sense to physicists, or anyone else for that matter. I mean the word slope doesn’t even have an “m” in it! But… I digress.
Now, I ask you to imagine that the same ice skating individual above stands atop a frozen ramp and is nudged ever so gently over the apex of the ramp so that they essentially start with no speed at all. You can hopefully recognize that our ice skater’s near zero speed will not remain that way. Instead they will hurtle down the icy incline with increasingly faster and faster speed. Well, if that is the case then for each second that they travel they will cover more and more distance along the ramp. Maybe after 1 second they are at 1 m along the ramp, but after 2 seconds they traveled 4 m, after 3 seconds they are 9 m from the start, etc.
The relationship in this case is known as a quadratic or a squared function. These are graphs that have the infamous shape of a parabola. And again in math class it would have been familiar to see y = ax2 + bx + c where SURPRISE “x” and “y” are variables again and a, b, and c are numerical coefficients or numbers. Ultimately, if your graph resembles a parabola, then it is a quadratic. In our ice skater example, final position (or x) and time (t) would be our variables and our coefficients would follow a = ½ the acceleration, b = initial velocity, and c = the object’s starting position. Of course, the same math rules apply. Like, you can only solve for the unknown time using the quadratic formula if you set the “y” equal to zero, or the final position in our case.
Another graph I would like to draw your attention to is an inverse relationship. As “x” goes up, “y” goes down. You have probably seen this written as y = 1/x. My suggestion is to always sketch out the function using a few points and you can quickly get an idea of its shape to compare it. For example, if we input x = 0, then y = undefined. If we input x = 1, then y = 1. Now things get more interesting, if you input x = 2, you get y = ½ so, increasing x causes y to decrease. That is an inverse relationship. Ultimately, your back of the napkin sketch will look a bit like the letter “L” but with a bit of curvature instead of the 90 degree turn located in quadrant 1 and 3 of the graph. For the sake of simplifying our understanding we will be mostly concerned only with the quadrant 1 space.
Finally, the last parent function you are likely to see is similar to the previous. It is known as an inverse-square function. In math class you would have seen y = 1/x2 and this relationship has quite a bit of relevance in our understanding of field forces. In fact, both gravitational and electric force exhibit inverse- square relationships. To better explain, I would suggest another quick sketch. If X = 0, the Y is undefined; if X = 1, y = 1/1; and X = 2, Y = ¼; while X = 3, Y = 1/9. The graph looks very similar to the inverse relationship explored earlier, but even more like an “L”. Now there would still be some curvature, just not as shallow of a curve as with the 1/x relationship.
Let’s shift gears a bit and talk about a problem solving scenario. In physics, most problems you encounter come in a word problem format. Now, that’s good because I have never encountered any problem worth solving in the “real world” that tells me what Y and X are. Once you read the problem, I strongly suggest drawing a picture that diagrams what is known about the object or objects involved. It helps to use short yet descriptive subscripts to differentiate values. Another useful practice is to list the Given information starting with those descriptive variables. Identify which physics principle is integral to solving the problem, usually that is as easy as looking for what is being asked, but it isn’t always so clear. Finally, plug in any values you have, and compute the mathematics ensuring that proper units are being maintained. Lastly, if you get stuck, see if you can describe the problem a different way. For example, it can help to make a graph focusing on the end point conditions. Sometimes seeing the graph can be enough to quickly answer.
To recap……
Math is an integral part of physics, and variables are related in numerous ways called parent functions. The four that you are required to be familiar with in AP Physics 1 are linear, quadratic, inverse, and inverse-square. Go into every word problem with a plan, and DO NOT FREAK OUT! If it feels overwhelming, draw a picture and list the given information. That is usually enough to get you started.
Coming up next on the APsolute RecAP Physics 1 Edition: One dimensional motion basics: displacement, velocity, and acceleration.
Today’s Question of the day is about Mathematic Parent Functions.
Question: Newton’s 2nd Law is often written F = m*a. If the force on an object remains constant, and the mass is doubled, what happens to the acceleration?