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A car is behind a truck going 25 $\mathrm{m} / \mathrm{s}$ on the highway. The driver looks for an opportunity to pass, guessing that his car can accelerate at 1.0 $\mathrm{m} / \mathrm{s}^{2}$ , and he gauges that he has to cover the $20-\mathrm{m}$ length of the truck, plus $10-\mathrm{m}$ clear room at the rear of the truck and 10 $\mathrm{m}$ more at the front of it. In the oncoming lane, he sees a car approaching, probably also traveling at 25 $\mathrm{m} / \mathrm{s}$ . He estimates that the car is about 400 $\mathrm{m}$ away. Should he attempt the pass? Give details.

The car should not pass.

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

Simon Fraser University

Hope College

McMaster University

Lectures

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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.

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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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So in this problem, we're trying to figure out if a car can pass a truck safely. So I'm just going to draw a situation we have right here. This is gonna call car one, and it's traveling at 25 meters per second. And in front of it is a truck, the capital T Uh, and they're separated by 10 meters. The truck itself is 20 meters, and then the car also wants to clear the truck by an additional 10 meters, and then we have a car coming in the opposite direction. Cartu And this Well, the speed is 25 minutes per second, but it's going in the opposite direction. So we'll say the velocity for this car is negative 25 meters per second, and then the car separated from the other one by 400 meters. So if this car one can accelerate at one meter per second squared, is it going to be able to pass the truck safely? So first I want to figure out how long it will take for this car to go around the truck and and to 10 meters in front of it. So for that first I know that for the car. We know our velocity is 25 meters per second. Same for the truck and the acceleration for the car. It's one meter per second squared and the trucks just moving on a constant velocity. And now the car. We know the car has to move, however far the truck moves, plus an additional 40 meters for this distance. So Delta X of the car is going to be Delta X. If the truck plus 40 meters and that should be enough information. So we know Delta X with the car, it's gonna be given by the initial velocity times, time plus 1/2 A T squared and Delta X for the truck just going to be its velocity. Time is time. This is philosophy and then because of this, we know that tells exit the car which is represented by this it's going to be the velocity of the truck times time, plus an additional 40 meters. Now we can solve this equation for time. So we know the initial velocity of the car is 25 and our acceleration is one truck is also going at 25 meters per second, so this means that we can cancel out some terms, and now we just get t squared equals 80. So t equals Route 80 which is about 8.94 seconds. So that's how long it takes the car to pass the truck. Now let's see how far the car went. Well, we already know from appear how far the car went, okay? Or at least an expression for it. It's going to be we'll see B zero still 25. But now we know our time is 8.94 seconds and then plus 1/2 times. Well, times one, uh, times now we have t squared, and this gives us adults of ex of 263.46 meters. And so that 263.46 is gonna be this distance. However, for the car guess. And so that means this distance on the right of the 400 meters. It's how far Car two is gonna travel or how far it can before it's not safe to pass. So this distance over here is 400 uh, minus where we got down here to 63.46 which is 136.54 meters. So now as long as this car car too travelled less than 136.54 meters in the time that it took the first car to pass that it will be safe to pass. So we go back up here the Delta X for card, too. It's the same expression as we did for the truck. It's just our velocity times time and now we know our time. So it's gonna be negative 25 meters per second times 8.94 meters. Okay, this is going to give us 223 223 and 1/2 meters. Well, negative is the displacement and that just means the car to was going left. So if Cartu is going 223 and 1/2 meters to the left and it only has 136 and change to make for car one to make safe passage, then we know that car one will not be able to pass the truck in time, and so it's not safe

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