The crash cushion for a highway barrier consists of a nest...

The crash cushion for a highway barrier consists of a nest of barrels filled with an impact-absorbing material. The barrier stopping force is measured versus the vehicle penetration into the barrier. Determine the distance a car having a weight of 4000 lb will penetrate the barrier if it is originally traveling at $55 \mathrm{ft} / \mathrm{s}$ when it strikes the first barrel.

Key Concepts

Kinetic Energy

Kinetic energy is the energy that a moving object possesses, calculated as one-half the mass times the square of its velocity. It is the energy that needs to be dissipated in a collision to bring the vehicle to a stop, and understanding this concept is crucial for analyzing how impacts are absorbed by safety devices.

Work-Energy Principle

The work-energy principle states that the net work done on an object is equal to its change in kinetic energy. In collision analysis, this means that the work performed by the stopping force of a barrier—through displacement—must equal the vehicle's initial kinetic energy, allowing one to determine the penetration distance required to halt the vehicle.

Integration of Force over Displacement

When the force exerted by a barrier varies with the distance over which it acts, integration of the force function with respect to displacement is used to calculate the total work done. This concept is essential in scenarios where the force is not constant, enabling precise determination of how much energy is absorbed over a given distance during an impact.

Transcript

00:01 Hi everybody, thanks for joining me today.

00:02 We're going to be looking at a problem that explores a relationship between kinetic energy and work.

00:07 So what we're given is we have a car that is moving uncontrollably at a speed of 65 miles an hour towards an array of steel barrels, a, b, and c.

00:20 And each set of barrels provides a different amount of force over a different distance.

00:26 So the barrels at a provide 18 kips from 0 to 5 feet.

00:30 The barrels at b provide 27 kips from 5 to 14 feet, and the barrels at c provide 36 kips from 14 to an undetermined.

00:41 What we're asked to solve for and what we're given is the mass of the car is 2 ,250 pounds.

00:48 The velocity of the car is 65 miles an hour, and we're asked to find what is the total stopping distance and what is the maximum acceleration of the car during its period of hitting those barrels.

00:59 So the first thing that we're going to do is we're going to actually assume.

01:02 That the car stops at less than 14 feet so we're only going to be using the forces from the 18 kips and the 27 kips and we're just going to use the again the equation that really is kinetic energy to work and we know that kinetic energy at the beginning is simply one half m v squared kinetic energy at the end since the vehicle is being brought to rest is zero and the one work is simply going to be those forces that i mentioned above acting over the prescribed distances.

01:40 So we assume that the car will be stopping at less than 14 feet.

01:45 So we know that the first set of the work done is due to the 18 kip force that's acting from 0 to 5 feet.

01:53 So that's going to be 18.

01:55 I'm just going to cite that as f .a.

01:57 Times xa...

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