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Many cars have $" 5 \mathrm { mi } / \mathrm { h } ( 8 \mathrm { km } / \mathrm { h } )$ bumpers" that are designed to compress and rebound elastically without any physical damage at speeds below 8$\mathrm { km } / \mathrm { h }$ . If the material of the bumpers permanently deforms after a compression of $1.5 \mathrm { cm } ,$ but remains like an elastic spring up to that point, what must be the effective spring constant of the bumper material, assuming the car has a mass of 1050$\mathrm { kg }$ and is tested by ramming into a solid wall?

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$2 \times 10^{7} \mathrm{N} / \mathrm{m}$

Physics 101 Mechanics

Chapter 7

Work and Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Rutgers, The State University of New Jersey

University of Washington

Hope College

Lectures

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

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In physics, the kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest. The kinetic energy of a rotating object is the sum of the kinetic energies of the object's parts.

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7.83 is about five mile per hour bumpers on cars, uh, which basically are designed to compress and rebound elastic Lee without any damage if they run into something at five miles per hour or less. So like bumping into things and parking lots things like this. So we want to assume that if the material compresses by one and 1/2 centimeters, we want to find what the, uh, upto one and 1/2 centimeters if with before it, you know, has some damage that occurs. We want to find its effective spring constant, and we assume that the car has a massive 1050 kilograms and is just tested by driving it into a solid wall at five miles an hour. So the work that the bumper does is, ah, going to be the change in kinetic energy of the car. The kinetic energy is decreasing, so the bumper does negative work, and since we're modeling, it is a spring. This is the work the bumper will do, and then the change in kinetic energy. The final speed is zero because it stops and then the initial is the initial kinetic energy is 1/2 and he squared where this is going to be our five miles per hour but in meters per second. So the spring constant, the effective spring concert is gonna be the mass of the car. Times are five miles per hour squared, divided by our 1.5 centimeters squared and doing all the appropriate unit conversions and then putting the numbers in. They get that This is two times 10 to the seventh Newtons per meter.

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