Physics Problems: Motion & Constant Acceleration

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We're set. We have this master set of kinematics equations. We should be able to solve any problems with constant acceleration that you can really think about posing. Any basic kinematics problems where you tell initial conditions. Where is something and how fast is it going at the beginning and what's its constant acceleration? I should be able to use these equations to figure out really anything I want to ask about its motion. It should be really a very powerful set of equations, and it really is.
So, imagine that I wanted to test these things out. I want to check these equations, so I take my old car and I go down to a local race track, abandoned for the day, so I think, "All right, I'm just going to make a whole bunch of measurements, positions and velocities and times and I'm just going to plug them in and see if the equations work." So I go down there. I'm sitting in the car. I got a stopwatch in my hand. And I start at the starting line and I see, good luck, there's a series of lines on the race track that are used to guide the drivers and the first one is at 25 meters. So, I think, "Cool, that's going to be my first checkpoint." So, I get in the car, jam on the gas, hopefully it's constant acceleration, and start the clock. So, I'm accelerating down the track and I pass the 25-meter mark and I click the stopwatch. It's five seconds and I'm looking at the road and I think, "Oh no, I blew it. I was looking at the stopwatch, I wasn't looking at the speedometer. I don't know how fast I was going, so I'm not going to be able to plug in the numbers and check."
So, I go back, kind of disappointed. I have to do it again. My poor old car can't do this too many times. And I think, "You know, I know these equations are correct. Why don't I use the equations? That's what they're for. Why don't I use the equations and figure out what I forgot to measure." I knew the following information. Technically speaking, I suppose I didn't really know either or xfinal. It's arbitrary. I could decide what I want to call the starting point. What I really measured was 25 meters is -. Let's just pick a coordinate system where we call =0, is thus 25, so I know that is 25 meters, is zero, and Dt in this case, the time on my stopwatch, was 5 seconds. And that's it. That's all the information that I got. It doesn't seem like enough to really do much with these equations.
But what I really wanted to do when I crossed that 25-meter line was to look at the clock and look at my speedometer. Can I figure out the velocity? So, that's the first question. What was my velocity when I crossed this line? So, it's got to be here, right? These equations contain all information about the motion, including the velocity as a function of time. So, I say, "Okay, I'll just use this one. =, uh-oh, I'm in trouble." And this is a common phenomenon. You're given a word problem in physics, you think you got enough information, you look at the equations, and you say, "Well, wait a minute, what am I going to do? I don't know the acceleration of my car yet. I do know the time. I know the initial velocity. I can't use this equation to solve for what I am interested in, the velocity when I crossed that line." So, I say, "Well, I'm okay. I've got a bunch of equations. How about this one?" I'm looking for at some moment. I know that I started at rest. Nope, no good. It's got the acceleration in it. So, I'm starting to get demoralized. In fact, the rest of the equations, well, these are the ones that had a on the left hand side and the question asked for . Look, when this happens to you, it's in there, okay? Don't panic. Just stop and think. It's a puzzle. Look at the last equation. I know Dx, is zero. Well, no, now I'm really getting nervous because it doesn't tell me , but ah-ha, if I look at this equation, I know this and this and this, since this is zero, Dt doesn't even matter. I know what Dt is. It's five seconds. I know everything here except for acceleration. I can find acceleration. I can calculate it. Now, once I know acceleration, I'm all set. I can use either one of these equations. So, let's do it. Let's plug in some numbers. I've got 25 meters, that's =+(Dt), which is five seconds, but I don't really have to plug it in if I'm multiplying it by zero. Plus one-half times times Dt quantity squared and Dt is five seconds. So, I just square that. And I can solve this equation. It's only got one unknown.
So, let's figure out what the acceleration is. I have to figure out acceleration, multiply both sides - oh, it's just a little algebra puzzle. Multiply both sides by 2, that gives me 50 meters. Divide by 5 seconds, quantity squared, and that's supposed to be the acceleration. Let's see what I got. 505², 5025 is 2 meters upstairs and second squared downstairs. Bingo. I've got the acceleration and now I can use either one of the other two equations. For instance, =+Dt. is what I'm after. It's zero plus two meters per second squared and what do I have to multiply that by? Well, I have to multiply it by Dt was 5 seconds. But 2m/s²x5 seconds is 10m/s²x seconds, so =10mps. So, I know and I know my acceleration. I've got another equation sitting there, which tells me what is. ² is ²+2Dt. Hah! I wanted to check the equations. Here's my opportunity. It's a second equation in which I can try to calculate . Let's see if I get the same answer. So ² is supposed to equal to ²+2. I know what that is now. 2m/s² times Dx. And Dx was 25 meters. So, 2x2x25 is 4x25, that's 100 and the units are meters squared per second squared. That's okay, because this is ². Take the square root of both sides. =10mps. That's the answer I just got. Checks, that's nice.
Now, stop for a second. is 10m/s. I've gotten it twice. I'm really confident that was my speed. I forgot to look at the speedometer, but I'm sure it was 10m/s, which by the way, is about 20 mph. Remember, way back we said velocity is distance divided by time. And I went 25 meters. And the time was 5 seconds. 255 is not 10. It's 5. So, what's going on? Distance divided by time, 255 is 5m/s. So, do I have reason to be worried here? Am I getting a different velocity, two different ways? No, it's okay. Distance divided by time, in this case, what I really mean here is DxDt of course. That is not instantaneous velocity. This is the instantaneous velocity when I cross the 25-meter mark line. This is the average velocity all the way from start to the 25-meter mark line. And, of course, it does make sense. The average velocity, if I started at zero and ended at 10; the average should be 5. So, actually everything is totally consistent and I'm all set. I've checked the equations. I've used equations. They're really - any problem that you have - sometimes it not immediately obvious. You look at them and the particular question that you were given doesn't sort of - maybe you aren't always just given and and Dt. That's kind of a preliminary kind of problem, but it's always there. Just a little puzzling and you can solve kinematics problems with constant acceleration.
Kinematics
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