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Chapter 2, Problem 2/166 MultistepCar A is traveling at the constant speed of 65km/h as it rounds the circular curve of 310-m radius and at theinstant represented i...

Question

Chapter 2, Problem 2/166 MultistepCar A is traveling at the constant speed of 65km/h as it rounds the circular curve of 310-m radius and at theinstant represented is at the position θ = 42°.Car B is traveling at the constant speed of 86km/h and passes the center of the circle at this same instant.Car A is located with respect tocar B by polarcoordinates r and θ with the pole movingwith B. For this instantdetermine vA/B andthe values of r˙ and θ˙ as measured by anobserver in car B.Find vA/B

Chapter 2, Problem 2/166 Multistep Car A is traveling at the constant speed of 65 km/h as it rounds the circular curve of 310-m radius and at the instant represented is at the position θ = 42°. Car B is traveling at the constant speed of 86 km/h and passes the center of the circle at this same instant. Car A is located with respect to car B by polar coordinates r and θ with the pole moving with B. For this instant determine vA/B and the values of r˙ and θ˙ as measured by an observer in car B. Find vA/B and vA/B. Answers: vA/B = (i + j) m/s vA/B = m/s



Answers

A car drives at a constant speed around a banked circular
track with a diameter of 127 $\mathrm{m}$ . The motion of the car can
be described in a coordinate system with its origin at the
center of the circle. At a particular instant the car's accel-
eration in the horizontal plane is given by
$$\vec{\mathbf{a}}=(-15.7 \hat{\mathbf{i}}-23.2 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^{2}$$
(a) What is the car's speed? (b) Where $(x$ and $y)$ is the car
at this instant?

Solving party of this problem. So here's radius is equal to 50 m and circumference of the circular party is equal to two. Pi R is equal to two pi multiplication. 50 m is equal to 314 metre now 14 top total circumference is equal to 314 by four metre is equal to 78.5 m. Now we all know that V f squared minus V I squared is equal to X So I will use this expression to calculate the value of VF is equal to under hood two x So just putting the value I can write we have is equal to root and the to multiplication to multiplication 78.5 So I get the value of VF approximately equal to 17.7 m per second. This is the answer for party now for part B, The radial acceleration e r is equal to b squared by our so just putting the value I can write 17.7 m per second holy square by 15 m So I get 6.28 m per second square now for party the total acceleration is equal to root under it is quite rural areas square, so just putting the value I can I do it squared plus 6.28 square on simplification. I get the value edge. Just look at it carefully. I get the value. Age 6.59 6.59 m per second, squared at an angle of 17.7 degrees east of south east of south. So this is our final answer.

For this problem we have to analyze Occur that this initially moving rest the surgical her But with the Costin tangential acceleration you called to to one single square circle. When then they circled the radius of the circular by two midterms. When were we need to find have to traveling in one working corpulence. We want to give it this circumference. So circle. Now we need to know what is the spirit of the car Now what is three Radial? A simulation of the car. But the broken its innovation off. So we can just start small a little. Well, car way have contention. Deceleration. We have 50 meters great news. And we can just gonna start by using the But first of all, we can No, But instead circumference circle of pop. Which one of you just good bye. I'm sorry. Stood by sequel to three 16 So who want to know? One right on the Eastern Conference she quote being 14 one suits by the by four Just sequel soon seven game Which to call? Okay, we can use the expression or yeah, square line Was the square the sequel to times assimilation? Thanks. We know that is then a X B courtesy that is the ark to the 1/4 circle Friends A quick listen. 54 since the cars stumping from breast the Mission velocity zero. And we're concerned for the velocity, the velocity. The last thing to be conflicting in this case is Arabia wasn't an issue, Stephen. Sort of things insurance innovation on the resistance, which is that's true if needed. The amount we find. Well, velocity with 17 7 We're sick now. The second part Look long. It was declined. Why stay, Rabia Incineration. So by with summation of horses, penitence, sickle loads Mars no longer a sequel to the square times. So lose, I was gonna be going to and with a screen well was and consuls because so great analysis information She's going to be one on seven square rainless freedom march on a stormy seas Me there Find me we can Since we already have green you I mention this intervention you can solve for their salt Broken assassination between Swirled Square Convention was prevention laws. We're listening still square los six mountain six square you could have in mind 6 57 Okay, square circum. And this is a little incineration from the car And this is a solution for the problem

Hello and welcome to this video solution of numerous. This question is based on the principles of circular motion. So here we have a car which is traveling east towards and then it turns to the not traveling in a circular path. And it's shown in the diagram. So let me draw this simple diagram. So the car is moving along a circular path. I draw this arc and initially it was traveling along the eastern direction. So like this and later on it moved towards is not And let's say, you know, at this point as the initial point as A. This B is somewhere around here and there is this point C. Towards is not now. This point B. There is another thing which is given here is that this point B makes an angle 35 degree from the horizontal. Now you have to calculate the acceleration of the car I'd be. So we need to calculate the acceleration and B. That is A. B. Now. Also it is given that the length of the Ark Abc is equal to look at 50 m. Now from here, we can actually calculate the radius of this circular part. Now this if you see is actually 1/4 of the total circumference rate. This arc, that means 1/4 time. The total circumference is two payard. That gives us 2 35. From here we can calculate are equal to 1 50 m. So the radius of the circular parties 1 50 m. Now. Also we can calculate the velocity of this car. So we have the total distance covered is to 35 m. We can actually calculate velocity like this. Let me show you. So, if you see what is the angular velocity omega, which is equal to the angular displacement of this car by that time taken. Right? So here the angular displacement is actually five way too. So it is covering 90 degrees. So I paid to weekend, Right? And the time taken his 36 seconds. So the angular velocities. Bye bye. To time statistics read per second. From here, we can actually calculate the angular acceleration of the car. That is the acceleration I'd be so a B. Did you see equal to when you got squared times out? So here we have omega. So let us calculate omega right here itself, which is equal to 0.0 436 Read par second. Right here we have 0.0 43 six squared times the radius is one with meters per second squared, which is equal to 0.28 0.285 meters per second squared. This is the acceleration of the car had been And this acceleration is actually directed toward the center of the circuit circular path. So here this is the yes, this is the answer to question. A So question is this. Now if you see there is a separate section of questions you have to do you know this acceleration in terms of vectors? Great. So we see that this acceleration A B is making an angle 35 degrees with the horizontal. Great. So considering that are for direction as G and to the right, this is why we can right. The component of A B along the horizontal direction will be okay, So let me write a B victor which is equal to 0.285 No, how long the horizontal direction? It will cost component rate minus cost 35 I and this is why the component will be positive rate. That means if you result, this component will like this. So this is why component that will have positivity. So that will have the sign component plus sign 35 Jacob and I. J. Jacob. Great. So we can directly calculate this as minus zero point two 33 Hi plus 0.163 She and the whole thing will be meters per second squared. So this is the victim notation of the acceleration at B. The next question is you have to german them. What is the velocity of the car? Oh sorry, I wanted speed of the car. So we're at speed of the God that we already got. Okay, well at speed with the car will be a sequel to the distance covered by the car. By the time taken here we have the total distance as this arc to 35. By the time taken is 36. That gives us the speed is 6.53 m per second. Next we have to calculate the average acceleration of this guy. So it is pretty simple option C the average acceleration is given by a beverage is given by us the final velocity minus initial velocity. Bye. That time. So here we have the final velocity directed towards the not rate. It will be plus 6.6 point 53 three G. Cap. And the initial velocity was explained. 53 plus 6.53 I. Sorry yes so this will be negative because here we have negatives of minus 6.53 Hi by total liberal by the time is 36. So here we have equal to minus one point I'm sorry. Zero point 0.181 Hi birth to your point 181 G. So this is the average acceleration of the car meter per second. I hope this is clear to you and have a very good rest of the day. Thank you.

Okay, So this problem is about a car traveling in a circular path and it's shown in the figure I'm gonna stop and look at the figure, okay? And we know that the length of the Ark ABC is 2 35 and it takes 36 seconds to make the turn. So then we have a few questions. First, let's get down the givens, and then we can discuss, actually what the, um, question is so I'll call the Ark linked to this. That's what I remember calling it. Um, when I took some math class, maybe remember the last time I did that, uh, to 35 meters. And the time was 36 seconds. Okay. And let's draw the picture. So, um, ABC and then be it's 35. It's do that a different color. So a is here. B is here, and C is here. And then we know that at B, this angle is 35 degrees in a double truck and great. So our first goal is to get the acceleration when the car is at the angle of 35 degrees and we want it in the unit in the terms of the unit vectors I hot and J a happ. So, um, if the acceleration at this point is going to be a centripetal because it's moving in the path of a circle and it's going to be the velocity squared divided by the radius But we don't know the velocity. What? We can figure it out using this arc length information. So, um so arc length in general, um, is equal to our times data. So, um, this state I would not be this angle, but the full angle that it goes through to go from a to C so that's gonna be equal to, um So I'll write that down. That data is pi over two. We'll call this 1 35 degrees Fi. I'm guessing we'll use it later, and yeah, and then we'll call this Radius R So, um And then, of course, it's, uh the centripetal acceleration is going to be towards the center, so that's where that 35 degrees will come in. But first, let's find the magnitude of it. So us is. Are they, uh, we know if they ah, are we given the radius? So we're not given the radius? I don't think just triple check. Okay. Well, we'll see. So So we have we have our data here. Um, just actually that. And we know that, um, Omega is Let's assume it's constant. It's gonna be the change in data over the change in time. So, um and then we also know Omega is equal to be over our. So now we have a direct connection to this. Be so we can say V is equal to R divided by time times data, and then we know asked is equal to arth data. So you can say asked over time, I guess I would have I was probably simply really think about that. The velocity is just a total distance divided by the total time. But that's okay. You don't have to always be as efficient as possible. Okay, so we know the velocity, and it's, um so we can put that in here. We have s square and divided by t squared times are and you know, we still have this are swim me pause and think how to deal with that. Oh, I see how we can deal with it. Um, looking at this, we actually know ass and then we know State s. So we confined Our our is just equal to s a birth data. Um, so we can go ahead and plug that in right here. So let's do that. So we still have the s squared. We still have the t squared. And then now we plug in Just plain are we're gonna have an ass here and a data here. So we have us data. Cancel one factor of us s data divided by a T Square. And I wonder if there was an easier way to get that. Not Not sure. So now we can start plugging everything in so to 35 times high over two. And we wanted to buy that whole thing by t squared. So 36 squared. And when I put that into a calculator, I got point to a five meters per second squared. Now that's in this direction. So, like, our acceleration vector looks like this. And then if this angle between here and here is 35 um, that's the same thing. Is this angle And so we can say than that the a vector. We could break it into its components. Its point needs 285 and then the horizontal component is in the to the left. By convention, we would say that's negative. And so it's gonna be minus co sign of 35 degrees in the eye hot direction. And then the vertical is gonna be a plus the sign of 35 in the J hat direction and so weaken, distribute that through. So let me go ahead and play got into my calculator. So I'll take that and co sign 35 degree is. And I got 0.233 negative, 0.233 and the I had direction and and then in the other direction we got, we wanna do sign. And then So we got 0.163 in the J hot direction. And then these both have units of meters per second squared. So I'll just put me there for a second square. Did you know that? Okay, And then next we went the average speed solely. Will that be so Average speed is gonna be the distance divided by the time. Oh, that's funny. Maybe there is an easier rate appoint acceleration, cause I already kind of came up with B is us over tea. So that's just to 35. Divided by 36. So let's put that into a calculator. Yes. 2 35 divided row. Not vice six by 36. Yeah, I got 6.53 meters per second. And for a C um, we went the average acceleration during the time interval. So this really highlights that you might say, Well, this feeds not changing. How do we have an average acceleration and acceleration is the difference in the final and initial vector divided by the total time so that we have the total time is just gonna be this tea and 36. But let's think about our XLR velocity vectors. So, um, to our final velocity is ah, the F and we want to subtract our initial velocity. The initial velocity was to the right, So we want to, um we want to add the negative of it. So here is negative v initial. So we want to add these two vectors. Um, so if you put them like, sort of tip to tail, um, and they have the same magnitude, we're assuming it's not changing its speed. Then the vector sum looks something like this, and it's really just gonna be, um this just by, um, trinkets gonna be the square root of two times the magnitude of the velocity. So scored a two times be something we want to divide that by time. So let's put that into a calculator. Um, so my previous answer times the square root of two and then we want to divide that whole thing by, um, 36. So I get 0.256 That's interesting that it comes out different than this cancer. Oh, do we want, I guess we want it in components, Not just the magnitude of it. So we, um So, actually, I guess the components of it are, um you know, here's the X and Y component, So this is its magnitude. So I guess I can change that to magnitude. Um, you fix this statement so that it's true. Um, so that's equal to the magnitude of a and then, um, yeah, And so now, but we want the difference between these. So you basically want take each of these velocities and divided by the time. So the velocity was, um, 25 divided by the by 36. So I get 0.181 um, in the and they're both the same. So yeah, I just took Sorry if it wasn't clear I took this velocity and divided it by time, Um, and then the final. And so, um, I'm like, I think I was clear. I might have just said the whole thing many different times and it might have been confusing someone, like, restart my explanation just to make sure I'm as clear as possible. So here's our final velocity vector. Here is our initial velocity vector and here's are the final sorry, negative initial. And then here's the final minus V initial. We want to take this thing and divided by time to get the acceleration and then we want to do it and component for him. So each component is just gonna be the velocity this velocity divided by time. Um, because he's air equal because the speed is equal. So we'll get that the acceleration is equal to those to the left. By convention, that's negative. So 0.181 again, I just did the velocity divided by time. And that's and that I had direction. And then the final velocity is positive, which my convention or is up, which by convention, this positive. So that's what I get for the acceleration and just go double check my numbers and he sticks and units on there. Yeah, so I


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