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(II) A stone is thrown vertically upward with a speed of24.0 $\mathrm{m} / \mathrm{s} .(a)$ How fast is it moving when it reaches a heightof 13.0 $\mathrm{m} ?$ (b) How much time is required to reach thisheight? (c) Why are there two answers to $(b) ?$

a) 17.9 m/sb) $t=0.620 s, t=4.28 s$c)There are two times at which the object reaches that height

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

Simon Fraser University

Hope College

University of Winnipeg

McMaster University

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.

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all right. So we have a stone was thrown upward with an initial velocity of 24 meters per second. And the first part we want to know are at height 13 meters. What is the velocity of the stone? And because this height is 13 meters and we're starting at zero or Delta Y, it's just why, minus y zero, which is 13 minus zero, which is just 13. We also know that a equals minus trees always And so we're looking for V that we can use this equation and we know v zero. We know a and we know don't know why. So that works out. You can just plug in. B squared is 24 squared AA minus because minus G two times 9.8 times 13 meters. And so our B squared is 321.2 and that means RV is 17.9 meters per second. And this can actually be plus or minus because we're taking the square root. And that answer that will be answered in the second part. Why it could be plus or minus. So for the second part, we want to know what time this happens. And because we're looking for tea, we want to use this equation. Delta. Why? It's zero t plus 1/2 a t squared. And I know that Delta, why is 13 meters easier? It was 24 a is minus G. And we have everything we need to sell for tea. But now we have a quadratic. So what I'm gonna do is put all the TI terms on one side actually gonna bring everything over to one side. So I get 1/2 g t squared minus 24 t plus 13 equals zero, and we'll plug in G uh, 4.9 p squared minus 24 t plus 13 equals zero. And if we saw this quadratic, uh, we actually get to solutions for tea, So t can either be 0.62 seconds or taken be, Ah, 4.28 seconds. And this is because if we're throwing a stone upwards is gonna go up and then it's gonna come back down, right? And at this certain height, we have 13 meters. I'll just call some height, 13 meters. We noticed this proble it's gonna hit that line twice, and so this is actually Ah, right here. It's the first tee and right here is the second T. And that's why we can actually have for V here. It could be plus or minus. The magnitude is always positive, but the direction could change because at this point, we have, uh, positive 17.9 meters per second. And at this point, we have negative 17.9 meters per second. And this is how we get to solutions for tea and actually two solutions your V as well.

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