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A woman is riding a Jet Ski at a speed of 26 m/s and notices a seawall straight ahead. The farthest she can lean the craft in order to make a turn is $22^{\circ} .$ This situation is like that of a car on a curve that is banked at an angle of $22^{\circ} .$ If she tries to make the turn without slowing down, what is the minimum distance from the seawall that she can begin making her turn and still avoid a crash?

170$m$

Physics 101 Mechanics

Chapter 5

Dynamics of Uniform Circular Motion

Newton's Laws of Motion

Applying Newton's Laws

Cornell University

University of Michigan - Ann Arbor

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.

03:43

In physics, dynamics is the branch of physics concerned with the study of forces and their effect on matter, commonly in the context of motion. In everyday usage, "dynamics" usually refers to a set of laws that describe the motion of bodies under the action of a system of forces. The motion of a body is described by its position and its velocity as the time value varies. The science of dynamics can be subdivided into, Dynamics of a rigid body, which deals with the motion of a rigid body in the frame of reference where it is considered to be a rigid body. Dynamics of a continuum, which deals with the motion of a continuous system, in the frame of reference where the system is considered to be a continuum.

03:35

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in this problem, we have to find the minimum distance that it will weaken, turn in order to avoid hitting a seawall. And we're told that we can approximate this problem like a banked curve problem. So if you use your picture, the one on jet skis as the red dot here and we draw asleep on a diagram, we know that our gravitational force right here must be pointing straight down, and our normal force is pointing perpendicular to our surface. When we draw more detailed diagram of our normal force, we can see that we're gonna be split that we were gonna be splitting it up into two components vertical component. But we're gonna call F y as well as a horizontal component effects. Now, since we're moving in a circular motion and FX is pointing toward the center of that circular motion, FX must be responsible for our centripetal force, which equals M. V squared over R. So, in order to solve this problem and find our radios, we're gonna have to find f of X. How to do that? We're going to look at our components so we can see from the problem. That data must be equal to this angle here as well. And therefore, since this black arrow is our normal force, that FX must be equal to F Um too. Nothin Sign data, but we don't know the value of s event. So we look to our diagram again and we can see that our why component must be canceling out the force of gravity and that that why component from the more detailed diagram must be equal to fn co sign of data. So now we have an equation for mg in terms of our knowns. So now we can write that are X component is equal to m g times the 10 of data by the fact that science data divided by Costa is equal to tension. So we can rewrite this as the squared over r equals G Tanith data. After canceling at her EMS, we can solve for our just v squared over g 10th ada and now all that have to do is plug in. The quantities that were given were told that the woman is traveling at 26 meters per second. We're going to square that quantity divided by the acceleration of gravity meters per second squared and then that multiplied by the art. The tangent of our angle. 22 degrees. And when we get from you sold for that, we get that our is equal to 170.73 meters, which is the final answer to our problem.

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