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The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion $ s = 2 \sin \pi t + 3 \cos \pi t $, where $ t $ is measured in seconds.

(a) Find the average velocity during each time period: (i) $ [1, 2] $ (ii) $ [1, 1.1] $ (iii) $ [1, 1.01] $ (iv) $ [1, 1.001] $(b) Estimate the instantaneous velocity of the particle when $ t =1 $.

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04:53

Daniel Jaimes

01:15

Carson Merrill

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 1

The Tangent and Velocity Problems

Limits

Derivatives

Sarah H.

September 23, 2020

How are you Doug? The answer for that is in physics, motion is the phenomenon in which an object changes its position over time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and time.

Doug F.

What is the term motion in physics?

Julia M.

I know this one! The average velocity of an object is its total displacement divided by the total time taken. In other words, it is the rate at which an object changes its position from one place to another. Average velocity is a vector quantity. The SI u

Eric V.

What is average velocity?

Erica G.

Hello there Brad as far as I know the physical velocity of a body in a point, or instantaneous velocity, is the velocity the body has at a specific time in a particular point of its trajectory. Instantaneous velocity, or simply velocity, is defined as the

Brad S.

What do you understand by instantaneous velocity?

Harvey Mudd College

University of Nottingham

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Okay let's take a look at this scenario. We have a position. Uh With time equation uh it is obviously we have oscillation. We have S. Is equal to two sine of pi T. Plus three. Cosine of pi T. And that's in centimeters and time is in seconds. And our goal is to find basically find the best approximation that we can't seem to say find, I'll say approximate. So we're going to approximate the instantaneous velocity At 1 2nd. That's our goal to get as close as we can. And we're gonna use that by taking a look at. Whereas velocity is the slope of the position graph the slope of the tangent line. We're gonna look at secret lines. We will get really close to uh where the two points almost become one. Therefore we'll get a good approximation. So we'll kind of hone in. So The first time period we're going to look at is one Between one and 2 seconds. And then we're gonna look at one and 1.1. See how the time period is getting shorter. And then we're going to look between one and 1.01. So that's all very short And then one and 1.001. Okay so those are the values we need to find. And so what we're gonna do is use the equation for the ob the ob is just the displacement over time. Just rise over run. So S. A. T minus S A one over t minus one and let's go ahead and plug in S. A. T. Is too sign up I. T. Plus three. Kassian up IT. And then we have to subtract off SF one. We know that when we plug in one that Trig functions have an argument. Just pie which is zero for a sign but one for well -1. Uh For Kassian right? Cosine of pi is minus one. So it looks like we'll get minus three. So double negative at three And that's over T -1. Okay so what we're going to do then is plug in our tea is just this it's the final value. Since we're always starting at one then the T. For this equation is just the T. Here. So let me show you how it works for the first one. You get the idea and we'll just plug in the numbers. Okay so if we are doing this first row we would say that the F. Is equal to two whoops to to sign uh two. We're plugging in two so two pi Plus three. Cosine 2 pi plus three. All over 2 -1. We'll sign a two pi is zero. Co signer to pi is one. This part is one. So we ended up getting um 6/1 or just six. So the average velocity is six cm/s. So these units should be by the way up here is centimeter per second. Okay so this is six. Alright if we instead of using two we plug in 1.1 for our tea. I'll just show you one more so I get the idea we'll start getting really ugly numbers. So VF is twice Sign of 1.15 plus three co sign of um Oh I forgot to double it. Let's go back and fix that. So let's go back our urologist erase and fix it. We have to do to ah Oh no no no no I got myself confused. We're fine. We're just plugging in T. And T is 1.1. So it's pi times t. 1.1 times pride. And same here at 1.15. Alright that's the argument for both sine and cosine and the underneath we'll put 1.1 -1. All right. So when we put this in the calculator we're going to start getting very very interesting decimal number. So here I'm going to write it on the way over here because -4.7120. If you replace the t by 1.01 You get -6.13411. If you're placed t by 1.001 You get -6.2 6837. And um notice at first we had big gas but as we get really close to one where the second point R T point is getting very close to one. Notice that we're kind of holding it around six and because we're doing tricks this kind of looks like you might think is that like to pie ish. And what we can do is if we just want to make sure is go ahead and do one where we really have a get really, really close. So let's say I have four zeros and then at once that's really, really close um I went ahead and did that earlier. When you plug that in you get minus um 6.28 three oh 37. So anyway, it's very very close to two pies. So I would say a very good guess for the uh the institute's velocity at one second Is about -2 pi That's a minus sign in front -2 pi centimeters per second. About the best guess we can make, but really very cool that we would get apply in our answer. So okay, anyway, hopefully that helped to have a fantastic day.

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