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(III) A lifeguard standing at the side of a swimming pool spots a child in distress, Fig. $53 .$ The lifeguard runs with average speed $v_{\mathrm{R}}$ along the pool's edge for a distance $x,$then jumps into the pool and swims with average speed $v_{\mathrm{S}}$ on a straight path to the child. (a) Show that the total time $t$ it takes the lifeguard to get to the child is given by $t=\frac{x}{v_{\mathrm{p}}}+\frac{\sqrt{D^{2}+(d-x)^{2}}}{v_{\mathrm{s}}}$(b) Assume $v_{\mathrm{R}}=4.0 \mathrm{m} / \mathrm{s}$ and $v_{\mathrm{S}}=1.5 \mathrm{m} / \mathrm{s}$ . Use a a graphing calculator or computer to plot $t$ vs. $x$ in part $(a)$ , and from this plot determine the optimal distance $x$ the life-guard should run before jumping into the pool (that is, find the value of $x$ that minimizes the time $t$ to get to the child).

a) $\frac{x}{v_{R}}+\frac{\sqrt{D^{2}+(d-x)^{2}}}{v_{s}}$b) $6.8 \mathrm{m}$

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

Suzanna K.

October 27, 2020

how to plot the graph?

Cornell University

University of Michigan - Ann Arbor

University of Washington

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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You must get from a point …

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Let one of the swimmer (sa…

05:28

Swimming The amount of tim…

01:06

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Solve each problem.Div…

So here. Let's redraw the diagram for the, um, for the ah, basically the pool. And we can say that if this would be the starting position of be the position of the drowning swimmer and then this would be the starting position of the lifeguard going straight down to the right. The lifeguard is going to walk some distance before jumping in the water. And then at one point, he's going to jump in the water and swim directly to the drowning swimmer. We know that this distance here is capital D being 8.0 meters, and we know that this distance here is lower case t being 10 0.0 meters. And so we can say that for each segment of the path this being Delta X, we can say that free segment of the path the time is given by the distance divided by the speed so that the time it takes to the lifeguard to reach the swimmer would be equal to the time it takes for the lifeguard to run on land, plus the time it takes to fill lifeguard to swim. Aah! This would be equal to the distance on land divided by the velocity on land The mother plus the distance in the pool divided by the distance Rather the velocity in the pool, the velocity which the life car can swim And so we can say that T is going to be equal to X divided by the velocity of how fast the lifeguard can run. So Visa Bar and then plus, we're gonna have to use Pythagorean theorem And we can say that rather, we can create a triangle here. And we're trying to find this distance here this high pot in use essentially. So we can say that this would be a plus the square root of D squared ah, plus D minus X quantity squared. And then this would be divided by the velocity with which the lifeguard can swim. So Bisa pests. And so that would be how we would prove the time taken to get from the lifeguard Thio the time taken to get to the swimmer from the starting position of the lifeguard four part B. This would be the graph so we can model this. Ah, and we can say that in order to minimize the time to get to the drowning child, we need to take the minimum of this graph. And essentially it would occur approximately right here. So we're going to see that this is 6.8. Therefore, team men, bikers at X, equaling 6.8 meters. Ah, which means that essentially, when essentially you should the the lifeguard should run 86.8 meters before before jumping into the pool in order to minimize the time taken in order to get the child. So this would be our answer for part B. That is the end of the solution. Thank you for watching.

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