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You are asked to design spring bumpers for the walls of a parking garage. A freely rolling 1200-kg var moving at 0.65m/s is to compress the spring no more than 0.090m before stopping. What should be the force constant of the spring? Assume that the spring has neglible mass.

In trying to solve this question I learned that the derivative of energy with respect to speed = force.

If you derivative: (1/2)mv^2 = mv.

Now suddenly mv = F, and ma = F.

My plan is to derivate (1/2)mv^2, get the force of the car, and say that the spring pushes with an equal and opposite force, and put the force into this formula:

Force = Force constant of spring/displacement

Force/displacement = constant of force which is the answer.

I tried to solve it with the speed formulas but I got an answer that is roughly the same, 200N apart. The answer to the question above is 63000N/m so I doubt my first way of doing it is the correct way.

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    $\begingroup$ Please note that Physics.StackExchange is not a homework help site. Please read this Meta post on asking homework-like questions and this Meta post for "check my work" problems. $\endgroup$ – Kyle Kanos Jul 31 '15 at 16:34
  • $\begingroup$ I am so tired of the moderators here. This is not homework. I am not doing homework in the middle of the summer. There is no school involved in what I am doing. I am studying on my own, and simply need some help. $\endgroup$ – David Lund Jul 31 '15 at 16:39
  • $\begingroup$ If you check the first link I provided, you would see that your question falls under the definition of homework that we employ here. $\endgroup$ – Kyle Kanos Jul 31 '15 at 16:43
  • $\begingroup$ "Providing an answer that doesn't help a student learn is not in the student's own best interest, and if a solution complete enough to be copied verbatim and handed in is given immediately, it will encourage more people to use the site as a free homework service. In the spirit of creating a lasting resource of mathematical knowledge, you may come back after a suitable amount of time and edit your response to include a more complete answer. Or even better, the student can post his own correct answer!" $\endgroup$ – David Lund Jul 31 '15 at 16:49
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    $\begingroup$ You've ignored the bit that reads This includes not just questions from actual homework assignments, but also self-study problems, puzzles, etc. The fact that you're not in school is irrelevant here. $\endgroup$ – Kyle Kanos Jul 31 '15 at 17:21
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This answer may not look like a typical answer, but I am attempting to instill a key concept, so please bear with me.

For this type of problem, where you are investigating a possible solution, UNITS are EXTREMELY important.

What are the units of momentum?

What are the units of force?

Note that if units do not match across an equal sign, the answer is guaranteed to be incorrect. If the units do match, your answer at least has a chance of being correct.

Further,

What are the units of kinetic energy?

What are the units for spring potential energy?

Have you tried conservation of energy, where the van's kinetic energy is converted into spring potential energy?

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  • $\begingroup$ I am not sure. I know that if you take ma you get force, so I'll think kgm/s^2. Kinetic energy is then 1/2mv^2, which must be .. 1/2*KG*(M/s)^2 or something. It's not before the next chapter that I am going to start with conversation of Energy, so I don't know. This chapter is named "Work and Kinetic Energy". It's weird have I have to do some hokus pokus on this problem when the other tasks have been very simple. $\endgroup$ – David Lund Jul 31 '15 at 16:23
  • $\begingroup$ When you deal with units, drop all of the constants. The "1/2" in your kinetic energy unit description doesn't belong. $\endgroup$ – David White Jul 31 '15 at 16:52
  • $\begingroup$ I will watch some videos youtube.com/watch?v=ZzwuHS9ldbY , so that I can probably use the method you guys are telling now. $\endgroup$ – David Lund Jul 31 '15 at 16:56
  • $\begingroup$ Hooke's law will give you the force on the spring vs. displacement of the spring. You have to integrate such a function to determine how much energy is stored in the spring. In other words, also study the chapter on conservation of energy before you attempt to finish your problem. $\endgroup$ – David White Jul 31 '15 at 17:01
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Well, it's not energy, its power $P$.

$~~~~~~~~~~~~P = \int F \cdot dv$

And since power is the derivative of energy $P = \dot E$, your world makes sense again ;).

Regarding your problem I agree with Yanping Cai, the kinetic energy of the car $E_{kin}$ must be converted into potential energy of the spring $E_{spring}$.

$~~~~~~~~~~~~\frac{1}{2} m_{max} \cdot v_{max}^2 = E_{kin,max} = E_{spring} = \frac{1}{2} k \cdot \Delta x_{max}^2$

$~~~~~~~~~~~~\rightarrow ~~ k = \cfrac{m_{max} \cdot v_{max}^2}{\Delta x_{max}^2}$

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I'm sure you can consider the force and integrate it over displacement to calculate the total work has been done.

But why don't you step back and consider the conservation of energy?

In short, $$\dfrac{1}{2}mv^{2} =\dfrac{1}{2}kx^{2}$$ $k$ is the minimum spring constant it requires.

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  • $\begingroup$ Welcome to Physics! Note that this site has MathJax enabled, which means you can use Latex-like syntax to add in equations for readability. $\endgroup$ – Kyle Kanos Jul 31 '15 at 15:54
  • $\begingroup$ Mr. Kanos - thanks for the hint regarding Latex. I'm currently not familiar with it, but I will start familiarizing myself with it immediately. $\endgroup$ – David White Jul 31 '15 at 16:12
  • $\begingroup$ I am going to watch a couple of videos about hookes to law to hopefully understand this. I am not familiar with energy conversions as of now. I am going through 1 chapter per 2 days, and I am only on chapter 6 of 10. Chapter 7 is where I start with Energy conversions. $\endgroup$ – David Lund Jul 31 '15 at 16:38

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