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Anderson Video - Car Accelerating Dragster

Professor Anderson
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 >> So car goes zero to 60 miles per hour in a time of 5 seconds. And let's ask the following question. What is the acceleration, what is that acceleration equal to? Okay, based on these numbers? Well, how would we figure this out? Yes, it's going to be kinematic equations. And let's take a look at this one. vf equals vi plus at. All right that's one of the kinematic equations. Is that going to help us figure this out? Maybe, we know this number right here right? v final, that's 60 miles per hour. We know this number right here, that is zero and we know time 5 seconds. and we know time 5 seconds. What we don't know is acceleration and that's exactly what we're looking for. So we can take that equation and we can solve for acceleration. We have vf equals vi plus at and now we can solve this for acceleration. We'll move vi over to the other side and we divide by t. Acceleration is vf minus vi over t. Now we know vi is going to be zero. That's going to drop out of our equation. vf is 60 miles per hour and t is 5 seconds. So first off, we have to get everything into si units. So 60 miles per hour, when I convert that to si units, all I have to do is multiply by 1. So let's get rid of the miles and then we'll get rid of the hours. So what do we know about miles and kilometers? What we might remember is there is 1.6 kilometers in a mile. And we of course remember that there are 10 to the 3 meters in a kilometer. All right, so we get rid of miles, we get rid of kilometers, we end up with meters. But you might also remember is that in 1 hour there are 3600 seconds. All right, 60 times 60 is 3600 and so now the hours will drop out. So let's run these numbers and see what we got, 60 times 1.6 times 10 to the 3 divided by 3600 and that's going to be meters per second. And if you guys have a calculator, put that in your calculator. Okay, so we approximated that as 30 meters per second, the exact answer was 26.67 meters per seconds which that sounds right because we're getting 60 miles per hour and we said it's roughly a factor of 2. Okay, so vf is that, let's plug it into our equation And see what we get for a. Okay we said, a was vf minus vi divided by t and we have 26.67, minus zero divided by t which we said was 5 seconds. So what is that? 26.67 over 5 is approximately 5. If you run it in your calculator, what do you get For the actual answer? >> 5.324 5 point what? >> 3 - 2 - 4. >> 5.3 meters per second squared. Okay that is the acceleration of your car, 5.3 meters per second squared which is about 1/2 g. So when somebody says how fast does your car accelerate, you can say 1/2 g right. g is 9.8, 5 is about 1/2 of that so you get about 1/2 g. Now, let's ask a follow up question. Let's say we're looking at a car that can accelerate a little bit faster than the average car. So we were just talking about your car running at about 1/2 g. But let's say you have instead a dragster. A dragster is all about acceleration. The dragster acceleration is on the order of 4 g, okay. It is really phenomenally quick. So 4 g is what? It's 4 times 9.8 meters per second squared. So it's on the order of 40 meters per second squared. It's roughly 8 times faster than a normal production car. So let's ask in the form of a question. How long will it take to travel 1000 feet if we start from rest? and then on a follow up we're going to ask what the final speed is. So, what do we do? We go to our kinematic equations. One of them we said is this: vx final equals vx initial plus a sub x times t. Is that going to help us get to the time? No not yet, because we don't know that. This we know. It starts from rest, that's zero. This we know is 4 g but that guy we don't know either. So we only have one equation and we have two unknowns and that's not going to help us do it. So instead let's go to this equation. x final equals x initial plus vx initial times t plus 1/2 a sub x t squared. And let's look at this equation and figure out what we know and what we don't know. We know this 1000 feet, we know this, that's zero, we know this, that's zero, we know this, acceleration. We don't know time. So we have one unknown in this equation so we can solve it for time. So let's just rewrite this and lets put in some of the zeros. Start at zero, start at rest, accelerate at 4 g and now we can solve this thing for t. You can double check my math, but I would say that we would get the following. 2xf over 4g square root is equal to t. 2xf over 4g square root is equal to t. So we have all those numbers. Now what we do need to change is xf feet and so we need to convert that to SI units. So 1000 feet if convert that in SI units I'd just multiply by 1 each time. I got 12 inches in a foot, we have 2.54 centimeters per inch and we have 1 meter equals 100 centimeters. And now you can put those in your calculator and double check with me but I got. >> 304.8. Good, 304.8 meters, right. 304.8 meters. So let's plug in and calculate what we get for t. There's our x sub f, let's plug it in this equation and see what we get for t. t equals the square root of 2 times x sub f, 304.8 meters and we're dividing by 4 times g which is 9.8 meters per second squared. And meters is going to cancel out, second squared ends up on top, take the square root and get seconds and if you plug it in your calculator and double check me, you should get 3.94 seconds. And this is how long it takes a dragster to start from rest and move 1000 feet. And it turns out 1000 feet is the real number that they use in the dragster. It used to be longer than that, it used to be 1/4 mile but the cars ended up going too fast and the tires would disintegrate and they didn't like that very much so they shortened the track a little bit, so the cars didn't end up going fast, as fast okay. We'll see how fast that is. What is the final speed of that dragster? Well now we can go back to our other equation, right. Our other equation was v final equals v initial plus a times t. And now we know all those numbers. v initial is zero. a we said is 4 times g, 9.8 meters per second squared. t is now 3.94 seconds. And if you plug in those numbers, we should get 154 meters per second. And if we remember our rule for going from meters per second to miles per hour, it's roughly double. So double this number and we end up with a final speed of around 300 miles per hour. And so now when you think about stuff like dragsters let's keep in mind what we're talking about. Your car was zero to 60 in 5 seconds. A dragster in 4 seconds goes zero to 300 miles per hour, okay. And these are real numbers. This is really what really happens at the drag strip, it is so phenomenal. Okay, hopefully that's clear. If not come see me during office hours. Cheers.