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A police car is traveling at a velocity of 18.0 $\mathrm{m} / \mathrm{s}$ due north, when a car zooms by at a constant velocity of 42.0 $\mathrm{m} / \mathrm{s}$ due north. After a reaction time of 0.800 $\mathrm{s}$ s the policeman begins to pursue the speeder with an acceleration of 5.00 $\mathrm{m} / \mathrm{s}^{2}$ . Including the reaction time, how long does it take for the police car to catch up with the speeder?

11.1$s$

Physics 101 Mechanics

Chapter 2

Kinematics in One Dimension

Motion Along a Straight Line

University of Michigan - Ann Arbor

University of Sheffield

McMaster University

Lectures

04:34

In physics, kinematics is the description of the motion of objects and systems in the frame of reference defined by the observer. An observer has to be specified, otherwise the term is meaningless.

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In mathematics, a position is a point in space. The concept is abstracted from physical space, in which a position is a location given by the coordinates of a point. In physics, the term is used to describe a family of quantities which describe the configuration of a physical system in a given state. The term is also used to describe the set of possible configurations of a system.

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this problem of police car is traveling at an initial velocity of 18 meters per second. When it sees a car going very fast, test him. It takes 0.8 seconds to react. So 0.8 seconds is the time that the police car takes to start to move or to move faster. And then after that moment it accelerates at a rate of five meters per second squared. And the question is, at what point in time will but police car catch this feeder. So let's define T as the reaction time plus the driver of the police cars acceleration time. So tea is the total time. So when we find our answer for tea, we'll have our full answer. Um, for the we'll start with the simple over here. Um, start with the car. The distance for the car would be velocity of the car text e. So we don't know what that distances, but we know it will be the same as the police cars distance. So we have an expression for the position of the car. So let's make an expression for the position of the police car There. The distance travelled of the police car. It's going to be the police cars, acceleration or or sorry, reaction, displacement, plus the acceleration, displacement. So what's happening while reacts? And then what happens when he accelerates? So the total would be the velocity times the reaction time and then for the acceleration part of the problem, we'll just use the not t plus 1/2 A T squared. Now again, we have to remember I'm gonna be specific here because we have to remember that it's not just tea, it's t acceleration, so I can actually plug it in that way. T acceleration plus 1/2 a t acceleration squared. But we said that the total time was the reaction time, plus the accelerating time so we could rewrite this here as t minus the reaction time. And we know that reaction time. So in just a second will plug a number. Okay, so we have the times, the reaction time, plus the initial velocity times the difference between the total time and the reaction time, plus 1/2 a times that same distant difference in Times Square. And that is an expression for the position of the police officer. But let's fill in some values so that we have a little bit more to work with. So are additional velocities. 18 The reaction time Waas 0.8. The initial velocities here is 18 t I don't know, but I do know the reaction time. 1/2 five meters per second squared times T minus 0.8 squared. So now I can simplify a little bit and eventually I'm gonna set of people to the position of the car. So 18 times 180.8 is 14.4 meters and then I'll distribute the 18. So 18 litres per second tens t minus 14.4 meters plus half of five is 2.5 and then I need to expand that by no meal. So, um, I get t squared minus 1.6 t plus 0.64 and now the second squared. So I've sort of dropped my units here in the middle, but I'll probably just drop all my units in the next line to make it easier. So let's do that. I have 14.4 meters minus 14.4 meters. That goes away so that I have 18 t plus 2.5, 2.5 t squared 2.5 times negative. 1.6 is negative for and then 2.5 times 0.64 is 1.6. So now I have a reasonable expression. I can actually combine my t terms, so I have to 0.5 t squared plus 14 t plus 1.6. And that's the position of my police car. So that is an expression making worker. So now we have the cars expression, which is 42 t and the police cars expression, which is, as I just said, 2.5 t square plus 14 t plus 6 1.6 So I sent them equal. 42 t is equal to 2.5 t squared plus 14 t plus 1.6 so we can get this Such that zero is on one side. We can use the quadratic equation, so let's subtract. 42 t from both sides will get zero equals 2.5 t squared, plus 14 minus 42 which is 28 negative plus 1.6. So if I used the quadratic equation, my A term is 2.5. My be term is negative. 28 my C term is 1.6. So that quadratic equation is negative. B plus or minus the square root of B squared, minus four a. C all over two. A. So I have positive 28 plus or minus the square root of 28 squared. That negative goes away when you square it minus four times 2.5 times 1.6, and that's all divided by two A, which is two times 2.5. So I have 28 plus or minus the square to 7 84 minus 16. You could do this on your calculator all in one step if you want. Over five 7 84 minus 16 A. 7 68 So I have 28 plus or minus the square root of 7 68 over five. The square root of 7 68 is just 27.7, so I have 28 plus or minus 27.7 over five. That gives me two answers. One. When you add the numerator, the terms of the numerator together in moment, you subtract. They both give the positive values, but one. When you add them, you get 11.1 seconds, and when you subtract him, you get 0.6 seconds. But if you look at this time, this is less than the reaction time. So this one doesn't make sense. He's not accelerating his total time. Can't be less than me. Um, reaction time. You'd have a negative acceleration time, so that doesn't make sense. So our only time that works is 11.1 seconds. And because we to find time to include the reaction time, this will be the total time that elapses before the police car catches this, Peter.

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