An energy-absorbing car bumper has a spring constant of 410 kN...

Question

An energy-absorbing car bumper has a spring constant of 410 kN/m. Find the maximum compression of the bumper if the car, with mass 1300 kg, collides with a wall at a speed of 2.0 m/s (approximately 5 mi/h ).

Accepted Answer

Hey, everyone in this problem, we're asked with the maximum compression of the bumper of a vehicle of mass 1400 kg equipped with an impact guard system having a spring constant of 420,000 newtons per meter. When the vehicle collides with a barrier at a speed of 3 m per second. As we're looking for the maximum compression. In this case, we have four answer choices all in meters. Option A 0.12 option B 0.22 option C 0.17 and option D 0.92. So in this case, OK, we have some speed, we have some compression in case we can think of this as some sort of spring. And so we want to think about is a conservation of mechanical energy. OK? We can assume that uh mechanical energy will be conserved. We have no um external forces. And so what we have is that the initial mechanical energy E Mac not will be equal to the final mechanical energy E MAC F. Now what makes up our mechanical energy? Well, it's gonna be kinetic energy plus potential energy. So we have knot, the initial kinetic energy plus U kno the initial potential energy is gonna be equal to KF plus EU F. And so the same values in that final case. Now when we have an initial case, in the final case, what we're gonna say is that the initial case is right before the collision when this vehicle is traveling at 3 m per second, and then the final case will be at maximum compression. Now, when we talk about potential energy here, we normally have gravitational potential energy as well as spring potential energy. In this case, gravitational potential energy is gonna be zero. And this is the vehicle staying at the same kind of vertical position. It's not moving up and down, there's gonna be no gravitational potential energy to worry about. So we have our kinetic energy given by one half MV squared. So one half MV not square in our elastic potential energy. So one half KX squared and we're using X knot squared to indicate that initial position and we can do the same on the right hand side, one half MVF squared plus one half KXF. Now, right off the bat, we can already eliminate some of these terms and simplify this. OK. At the beginning that spring is not being compressed, it's not being stretched, it's at equilibrium. So the X not value and that compression or stretch value is going to be zero and it is not being compressed or stretched. That distance is zero. Hey, no, spring potential energy at maximum compression. OK. It's going to come to rest momentarily before that spring starts to come back the other way. And so the final speed VF at maximum compression is going to be zero. So no kinetic energy at that final time. And we also have this one half term we can divide by. And when we do that, we're left with the math multiplied by the initial velocity squared or the initial speed squared is equal to that spring constant K multiplied by that final X position or X distance story. And we're gonna call this X max to make it really, really clear that that's that compression that were looking. OK. So let's go ahead and solve for X max. We can divide both sides by K take the square root, we get their X max is equal to the square root of MV, not squared divided by K going ahead and substituting in these values X max is equal to the square root of 1400 kg multiplied by 3 m per second squared, divided by 420,000 newtons per meter. You know in terms of units, a newton is equivalent to a kilogram meter per second squared. Nice the units of kilograms will con divide it, OK. One of these meter terms will divide it, the second squared will divide it. So we're gonna be left with inside the square root is a unit of meters. Squared. And when we take the square root, we get the unit of meters that we want. And when we work this out on our calculator, we get that this maximum compression X max is approximately equal to 0.1732 m. And that's exactly what we were looking for in this problem. If we go back to our answer choices, we round to two significant digits. We can see that the maximum compression of the bumper of this vehicle is gonna be option C 0.17 m. Thanks everyone for watching. I hope this video helped you in the next one.

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An energy-absorbing car bumper has a spring constant of 410 kN/m. Find the maximum compression of the bumper if the car, with mass 1300 kg, collides with a wall at a speed of 2.0 m/s (approximately 5 mi/h ).

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