FRQ #3 AP Physics 1 Linearization

Added: 2026-05-09

[00:00:01]Hi, good morning everybody. Wanted to do a quick a make-believe linearization problem.

[00:00:07]Um this one's looking at a cart that's starting at the top of a ramp, goes down the ramp, hits a spring, collides, and squishes it.

[00:00:15]And they have two pieces of data here.

[00:00:17]One is they're measuring the height they drop it from, and then they're measuring the squish. Both are in meters, so we don't have to convert it. And then up here it also tells us the cart's 4 kg.

[00:00:28]So, first thing I do is when I see this, I have to make a decision. Am I looking at uh um forces, energy, momentum? And so, it's one of those three, right?

[00:00:38]Um well, anytime I have a change in height, I'm going to I'm going to usually go think about energy.

[00:00:44]There's a collision with a spring, but the spring doesn't have mass, so I can't really do momentum.

[00:00:50]I don't want to do forces cuz the forces are changing. So, if you look down below, they actually say, "Hey, this is where this is our result. The student correctly came up with this." So, if this is an FRQ, there'd be a part before this where you came up with your own lab, but here um they're they're giving it to you, okay?

[00:01:09]So, they give you the data, they measure the heights, and they measure the squish.

[00:01:14]And then they said the student correctly comes up with an equation. And they do this a lot.

[00:01:19]That way you're not double dinged if you can't make an equation, they they work with you. So, the question they ask here is, "Indicate what measure or calculated quantity could be plotted on an axis to yield a linear graph whose slope can be used to calculate the value of the spring constant K?" So, here it goes. So, here's my goal.

[00:01:41]It's to get K.

[00:01:43]And then I'm going to look and see what did they give me in this problem? They gave me H. So, I see H right here.

[00:01:50]And they gave me X. I see X right here.

[00:01:53]Well, the H the H doesn't have any powers to it. It doesn't have a squared, it's not in a square root, it's just H.

[00:02:00]Look at the X though.

[00:02:01]The X is squared.

[00:02:04]So, um I didn't graph this, but if I were to just graph these two, what you end up seeing is a graph that looks like this.

[00:02:16]And you're like, "What?"

[00:02:18]Um that's not what I wanted.

[00:02:22]That is not a line. So, if I just graph those, that's what I get. Now, on the AP exam, you have Desmos, you could graph it real quick and see, "Oh, is it a line already?" Right? That's one option.

[00:02:32]The other is you can look I'm going to give you tools to look at the equations here. So, here we go. See how the H H is good. H is H is to the power of one, there's nothing on it. But you see the X how it's X squared?

[00:02:44]Since that's X squared, I'm not going to graph X, I'm going to graph X squared.

[00:02:52]And this is why they give you the extra column. If they don't give it to you, write it right next to it.

[00:02:56]I'm going to literally calculate X squared. So, I'm going to calculate X squared. So, 0.5 * 0.5, I get 0.25.

[00:03:06]And here I got 1 squared.

[00:03:09]And I got 1.5 squared.

[00:03:13]2.25.

[00:03:17]And 2 squared, and then 2.5.

[00:03:30]All right, 6.25.

[00:03:33]Okay?

[00:03:34]So, if I were to graph this, X squared and H, I'm going to get a line. So, down here, I'm going to tell you right now, in the X direction, um let's have Actually, it just however we solve this.

[00:03:51]So, I'm going to I'm going to say H is going to be in my Y, and X squared is going to be in my X.

[00:03:59]It's just what I felt like, it's what I'm going to do. So, I'm going to put H here, X squared here. So, the first thing I do is I put the units, meters. Here, X squared, since it's meters, I'm going to do meters squared.

[00:04:11]Okay?

[00:04:15]Now, I need to get out my ruler, and I'm going to label my axes.

[00:04:23]Okay? So, first thing you got my axes here.

[00:04:30]And my H's go from basically 0 to above 5. So, maybe go to 6. So, 1 2 3 4 5.

[00:04:40]Um go to 10, I'll only go halfway, so maybe I I could do that. 8 6 4 2 0.

[00:04:52]Not perfect, but it's okay. So, 5.1 will be over here, okay? And then my X, I'm going to go from basically 0 to 6. So, maybe you do the same. Oops, sorry. 0. I'm okay. 4 6 8 10.

[00:05:07]Right? Not perfect, but So, I'm at 5.1, I'm at 6.25.

[00:05:21]About here. Not perfect, but about there, okay?

[00:05:24]When I'm at 3.1 in H direction, 3.1, I'm at 4. So, 4 and 3.1.

[00:05:34]Okay?

[00:05:35]And then 1.7 is down here, so there's 1.

[00:05:40]1.7, 2.25.

[00:05:42]Kind of here.

[00:05:44]Not perfect, but right? Getting close.

[00:05:47]Um 0.9, so that's 1, it's just below 1.

[00:05:51]It's at 1.

[00:05:53]And 0.2 is at 0.25.

[00:06:00]If you look, now we're seeing does that thing look like a line? Yeah, it does.

[00:06:06]Okay, good job.

[00:06:07]All right. So, now, best fit line. So, let's go over rules of best fit line. It does not have to go through the first dot and the second and and first dot. First dot and last dot, it does not.

[00:06:17]Second, it it it needs to have about the same above as below.

[00:06:22]And so, I'm just going to try to draw a line.

[00:06:28]I mean, that that dot is kind of bothering me, but I got some below, I got some above.

[00:06:33]It doesn't have to go through them.

[00:06:35]There we go. There's my line.

[00:06:38]So, it doesn't have to go through them.

[00:06:39]Try to balance above and below, and draw a line. Don't draw multiple lines. Don't draw a squiggle, just draw one line.

[00:06:48]Now, to find the slope, mine conveniently goes through 0 0. Let's say it didn't, so that way I can practice with you here.

[00:06:55]So, I'm going to circle like, "Ooh, see how it goes through the bull's-eye right there?"

[00:06:59]Okay?

[00:06:59]And then here, ooh, it goes through that one.

[00:07:03]I like that one.

[00:07:06]Okay?

[00:07:07]So, if I want to find the slope of this, I'm going to look at rise over run.

[00:07:18]So, it rise. So, I got to figure what that number is. So, that's right here, and it goes right here.

[00:07:24]So, this is 2, this would be 3.

[00:07:28]So, this is That's 3. 1 2 and 1/2. Oh, jeez. My scaling is not very good on this one.

[00:07:36]So, let's see. This distance is 2.

[00:07:41]And I'm going to divide that by how many things I have, five. So, each line is worth 0.4.

[00:07:47]That's worth 0.4, that was worth 0.4, right? So, this is going to be 2.4, 2.8.

[00:07:55]So, this is at 2.8 right here.

[00:08:00]And this one goes up to 5.6. Cuz remember every line segment is going to be 0.4.

[00:08:06]So, vertically, it goes from 2.8 to 5.6.

[00:08:09]So, it goes vertically 2.8.

[00:08:12]Horizontally, it goes from 4 to 7.6.

[00:08:19]So, this one goes horizontally 3.6.

[00:08:23]So, my slope here is going to be rise over run.

[00:08:30]Now, we have no idea what this means right now. So, we kind of did this a little out of order, but I got a slope of 0.778.

[00:08:38]That's my slope, okay? We're going to probably use it later cuz it told me I need to use it, okay?

[00:08:44]So, notice, by graphing that and that, I ended up getting a straight line. Okay?

[00:08:52]So, notice, our points here looking at uh the exam, did I label my axes? Yes.

[00:08:57]Did I plot the points? Yes. Did I draw the best fit line? Yes.

[00:09:02]Part D, calculate the value of the spring constant. Now, we're down to the tough stuff, all right? So, >> [snorts] >> this is now going to be part D here.

[00:09:13]Well, I'm going to write out what I have here.

[00:09:16]I have 1/2 k x squared = mgh.

[00:09:23]Now, I made the vertical axis Y. So, I made this Y.

[00:09:30]Okay?

[00:09:31]And I made X squared I made that equal to the the X direction in my graph. So, I'm going to put X not like an X, but like like horizontal, okay?

[00:09:43]Remember in math class we had Y = MX + B?

[00:09:47]All right. So, notice how Y is by itself.

[00:09:50]So, see how this is my Y?

[00:09:53]All right. Try to parallel this to what we did above. That's my x squared, and that's my h.

[00:09:59]Well, I want that to be my y. I need it by itself.

[00:10:04]So, I'm going to divide both sides by mg.

[00:10:08]So, I'm just going to write down 1/2 kx squared.

[00:10:12]And this whole side is divided by mg.

[00:10:16]I think you'll do it that way. So, let me write it that way. Okay?

[00:10:19]So, notice do I have this is like my y by itself? Yep.

[00:10:23]See how my x squared?

[00:10:25]That's going to be my x.

[00:10:28]Yep.

[00:10:29]And then this, all of the rest of the stuff.

[00:10:33]See my equation here is y equals mx plus b?

[00:10:36]This whole thing right here is going to be my slope.

[00:10:43]So, this is my y.

[00:10:45]This is my x direction.

[00:10:48]And this is my slope. If I had a plus or minus here, that'd be my y intercept.

[00:10:53]That's what this stands for, y intercept. Okay?

[00:10:57]So, watch this. My slope will equal 1/2 k over mg.

[00:11:05]Well, I know what my slope is equal to.

[00:11:07]My slope is equal to.778.

[00:11:11]Half is 0.5.

[00:11:15]Okay? What's my m in this problem?

[00:11:19]Oh, we have here it says it's 4 kg.

[00:11:23]I have a bad feeling this isn't going to not to focus the entire time.

[00:11:27]Please don't yell at me. Um And so, the bottom is mg. So, that's 4.

[00:11:34]So, I got.778 equals.5 over 40. I'm going to simplify that. So,.5 over 40, I get 0.0125 k.

[00:11:46]I got k by itself, I divide. So, k equals 62 N/m.

[00:12:02]All right. I know this is a lot, but I just wanted to give you another resource um as we head into that AP test. Um you guys are doing great. Keep it up. Okay?

[00:12:10]Um let's see. I'll get a one last shot on here so you can see the whole page, but hopefully that helped a little bit.

[00:12:16]Remember you got AP dailies that deals with linearization, so go check those out, and just keep letting us know. All right. Keep it up, everybody.