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You are traveling at a constant speed $v_{\mathrm{M}},$ and there is a car in front of you traveling with a speed $v_{\mathrm{A}}$ . You notice that $v_{\mathrm{M}}>v_{\mathrm{A}},$ so you start slowing down with a constant acceleration $a$ when the distance between you and the other car is $x .$ What relationship between $a$ and $x$ determines whether or not you run into the car in front of you?

$x>\frac{\left(V_{M}-V_{A}\right)^{2}}{2 a}$

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

Cornell University

Simon Fraser University

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.

02:13

You're in a car trave…

01:00

You’re traveling in a car …

04:07

A car travelling at a spee…

00:54

A car has an initial veloc…

07:40

A car traveling at speed $…

09:55

You're speeding at $8…

03:25

After you apply the brakes…

01:28

Two cars with the same spe…

02:21

A car, of mass $m$, travel…

So you're in a car, going at a constant speed of the M. And there's a car in front of you also going the same direction with speed V A. And you want to slow down before you hit. You have Don't sit well. Actually, the Delta X is X in this case. So we want to know you want to be able to start from this speed V m and slow down until you hit the speed of the car in front of you and wants to match its speed. Then you know you're not gonna hit it. So we know our initial velocity is gonna be the velocity that we're currently going up on. Our final velocity is the goal. We want to hit the V A. Disputed the car in front of us and the Delta X we have to do that is X, and we're also given our acceleration. And what I interpret from that is that they're giving us the magnitude of the exploration. So we actually have to make it negative A to work with the problem. But that's just my interpretation. So we know we can use the equation with these four variables in it. VF squared is the initial squared plus two a adults X And now if we plug in, these numbers will get the A squared is B M squared minus two a x minus that here and now we want to find the relationship between A and X. So this equation is saying, uh, that this is this would be the exact if if we were to slow down from Vienna via and exactly X, and then we'll be touching the car in front of us. So we just get the velocities on both sides and then we can compare a an X ensues difference between those two is minus two a X and so to switch them around B m squared minus B. A squared is positive to a X, and then we'll divide by two a. The M squared minus b a squared over two a equals x, and this says that in exactly the distance X, we will be able to slow down. And so as long as X is greater than this distance, then we won't hit the car in front of us. But if it's less than this, then we will have the car So what we have to say for the inequality to compare Alien X, that's we want the distance separating the two cars to be greater than this expression on the right. And as long as we have enough separation between two cars, then according to this, we will be able to slow down in time before hitting the car in front of us.

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