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Spring Constant to Bring a Car to Rest

  1. May 7, 2015 #1
    1. The problem statement, all variables and given/known data
    What should be the spring constant k of a spring designed to bring a 1400 kg car to rest from a speed of 28 m/sec so that the occupants undergo a maximum acceleration of 5g’s?

    I have the solution manual and can see how they did this, but am curious as to why my attempt did not work. It didn't seem to break any rules that I can see.

    [itex] a = -5g = -49 {\frac{m}{s^{2}}}[/itex] neg. due to deceleration
    [itex] m = 1400 kg, v_{0} = 28 {\frac{m}{s}}, v = 0 {\frac{m}{s}}[/itex]

    2. Relevant equations
    I went about the problem in a roundabout way. The equations used are:
    [itex]v = v_{0} + at[/itex]
    [itex]x - x_{0} = v_{0}t + {\frac{1}{2}}at^{2}[/itex]
    [itex]KE = PE[/itex]
    [itex]{\frac{1}{2}}mv^{2} = {\frac{1}{2}}Kx^{2}[/itex]

    3. The attempt at a solution
    [itex]v = v_{0} + at[/itex]
    [itex]0 = 28 + -49t[/itex]
    [itex]t =0.57143 s[/itex]
    [itex]x - x_{0} = v_{0}t + {\frac{1}{2}}at^{2}[/itex]
    [itex]x - 0 = (28)(0.57143) + {\frac{1}{2}}(-49)(0.57143)^{2}[/itex]
    [itex]x = 8[/itex]
    [itex]{\frac{1}{2}}(1400)(28)^{2} = {\frac{1}{2}}K(8)^{2}[/itex]
    [itex]K=17150 Nm[/itex]

    I believe the correct answer is 4288 Nm based on a similar problem in the solution manual. I don't understand why working it like this does not work though. All the units work out, and while I know that's not a guarantee of correct setup it is usually a sign you are moving in the right direction. Anything stand out in particular here?

    Thanks!
     
  2. jcsd
  3. May 7, 2015 #2

    SammyS

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    Your answer is 4 times the correct answer.

    With your answer, what force does the spring exert at maximum compression ?
     
  4. May 7, 2015 #3
    Since the car travels 8 meters to stop given a constant deceleration of 5g wouldn't the max compression be 8m? Which would make the F=17150*8, a very large number..

    But we took time into account when we found the x distance, so I'm not understanding why the k constant is so large. It feels like it's giving me what k would be if it needed to decelerate the car at 1 specific moment instead of over a span of time... But I know that last kinetic energy equation is right as they are both conservative forces.
     
  5. May 7, 2015 #4

    SammyS

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    Take that force and divide it by the mass to find the acceleration, as the car comes to rest.
     
  6. May 7, 2015 #5
    Oh. So the deceleration cannot be constant if it is only a spring causing it then due to the increasing force as it compresses more? Is that why the basic kinetic equations fail here, because they're finding a constant acceleration?
     
  7. May 7, 2015 #6

    SammyS

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    Who is "they" ?

    I believe that your solution does make the average acceleration equal to -5g.
     
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