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A city planner is working on the redesign of a hilly portionof a city. An important consideration is how steep the roadscan be so that even low-powered cars can get up the hills without slowing down. A particular small car, with a mass of$920 \mathrm{kg},$ can accelerate on a level road from rest to 21 $\mathrm{m} / \mathrm{s}$$(75 \mathrm{km} / \mathrm{h})$ in 12.5 $\mathrm{s} .$ Using these data, calculate the maximumsteepness of a hill.

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the maximum steepness of the hill is $\left[9.9^{\circ}\right]$

Physics 101 Mechanics

Chapter 4

Dynamics: Newton's Laws of Motion

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

Moment, Impulse, and Collisions

University of Michigan - Ann Arbor

University of Washington

Simon Fraser University

University of Winnipeg

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.

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

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A city planner is working …

02:54

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01:11

A $1200-\mathrm{kg}$ car …

03:05

(a) How high a hill can a…

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(a) How high a hill can a …

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The mass of a car is $1500…

02:00

The mass of a car is 1500 …

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On mountainous downhill ro…

04:04

(III) A bicyclist can coas…

02:29

Review problem. The mass o…

04:01

An automobile weighing $12…

So here we need to calculate steepness of the hill. So we have the hill here we have the car here Well represented by a box perpendicular to the surface of contact will always be the force normal. Always going straight down would be the force of gravity. And then, um, going up the hill would be the force applied by the car. So we can say that force, Part VI final velocity final would be equal to a velocity initial plus 80. Ah, we know that here the initial velocity would be equal to zero. And so we can say that the acceleration would be equal to the final velocity divided by T. This would be equal to 21 meters per second, divided by 12.5 seconds. This is equaling 1.68 meters per second squared. And so we can say that the force applied for the car would be equal to M A. This would be equal to the mass of the car. 920 kilograms multiplied by the acceleration of 1.68 meters per second squared. And we find that the force applied there's gonna be equal to 1546 Newton's. Now, if we're going to assume that this is the force pushing the car on the incline as well, um, we can then, um, consider the free body diagram for the car climbing the hill. Ah, here. We're going to now apply Newton's second law to the car. So in doing that, we can say that the sum of forces in the X direction with the equal to force applied minus mg sinus data and this has to be equal to zero because the car is moving at a constant velocity because we don't we want the car to have a constant speed on the maximum incline. And now we're gonna solve for theta in order to solve with the greatest steepness. So Fada would then be equal to arc sine of force applied, divided by the weight of the car. So this would be arc sine of 1546 Newton's. This would be divided by 920 kilograms multiplied by 9.80 meters per second squared, and we find that the fate of would be 9.9 degrees. So anything greater than 9.9 degrees and the car is going to slip and gravity will take over and the car will start decelerating and eventually will slowly, uh, travel down the hill. So our maximum steepness is again 9.9 degrees. That is the end of the solution. Thank you for watching.

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