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(III) The acceleration of an object (in $\mathrm{m} / \mathrm{s}^{2} )$ is measured at 1.00 -s intervals starting at $t=0$ to be as follows: $1.25,1.58$$1.96,2.40,2.66,2.70,2.74,2.72,2.60,2.30,2.04,1.76,1.41,1.09$$0.86,0.51,0.28,0.10 .$ Use numerical integration (see Section 9 of Describing Motion: Kinematics in One Dimension) to estimate $(a)$ the velocity (assume that $v=0$ at $t=0 )$ and(b) the displacement at $t=17.00 \mathrm{s}$

a) $30.3 \mathrm{m} / \mathrm{s}$b) $305 \mathrm{m}$

Physics 101 Mechanics

Chapter 2

Describing Motion: Kinematics in One Dimension

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

University of Washington

Hope College

Lectures

03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

04:16

In mathematics, a proof is a sequence of statements given to explain how a conclusion is derived from premises known or assumed to be true. The proof attempts to demonstrate that the conclusion is a logical consequence of the premises, and is one of the most important goals of mathematics.

06:04

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So this question is about numerical integration, and I'm just gonna do all this on an Excel sheet and truly the method. But doing every single calculation takes a long time. But if you want to do it by him, go ahead. I'm just going to show you the quick way. So for each of these, what were given time and acceleration values for the 1st 17 seconds and we know that velocity is gonna be the integral of the acceleration and the position is going to be integral of the velocity and in order to find those So we know the starting velocity and position is zero meant to get this first value. I'm gonna take the basically the average of the last two accelerations, so some over two or times 1/2 and then multiply that by the time interval over those two accelerations. And for all of these, the time interval is just one second. And then for each velocity, I'm gonna add in the velocity of the one before it. And so for the starting one, it's just zero plus Ah, the average of the explorations. And you keep doing that down and you go all the way down until you get to ah t equals 17 for the time and you get Ah, the velocity is 30.285 meters per second and the only one to get this is you have to do every single value individually and some of them all up. You can't just go straight to the end value and expect to get an answer. Ah, and the same thing is true for position. So what I do is take the average of the last two velocities, multiply it by the time interval. So the difference of the two times again one second for every interval and then add in the previous position. And if you do this all the way down Ah, in order to do the positions, you also need every single velocity value. So you have to do all these out. And if you do that down, then you get three or 4.872 meters is the position at 17 seconds. And doing this with Excel is really, really quick and easy because you just plug in one formula and then it works for all the rest. But doing it by hand is a pain. Um, again, it would take me a while. The show it every single calculation, but the way you're probably going to be doing this, if you ever do numerical integration is using Excel for data analysis. So this is just what you do. Um, you have to step through every value and keep adding them up. And that's basically how integration works, but not analytical its numerical this time.

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