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Acceleration in One Dimensional Inelastic Collision

  1. Jul 10, 2017 #1
    1. The problem statement, all variables and given/known data
    Two cars of 540 kg and 1400 kg collide head on while moving 80 km/h in opposite directions. After the collision, the automobiles remain locked together.
    Find the velocity of the wreck, the kinetic energy of the two-automobile system before and after the collision, and find the acceleration of the passenger compartment of each vehicle given that the front end of each car crumpled by 0.6 m during the collision.

    (I've already found the velocity and energies, I just need help with the acceleration.)

    2. Relevant equations
    Kinematic equations?

    3. The attempt at a solution
    a(x-x0) = (1/2)(v2 - v02)
  2. jcsd
  3. Jul 10, 2017 #2
    They are asking for the acceleration, but what they really want is the average acceleration during the collision. The equation you wrote is the relationship they expect you to use.
  4. Jul 10, 2017 #3
    Maybe I'm doing something really wrong with this equation, but I'm not getting anywhere near the right answer (-130 m/s, 850 m/s). I keep getting -330 m/s using the final velocity I calculated in the first part, which is 9.8 m/s in the direction of the 1400 kg car.
  5. Jul 10, 2017 #4
    Let's see the details of your calculation for the velocity of the wreck. I get a wreck velocity of <10 m/s.
    Last edited: Jul 10, 2017
  6. Jul 10, 2017 #5

    Sorry for the lack of LaTeX. The app doesn't support it.

    9.8 m/s is the same value the book gives.
  7. Jul 10, 2017 #6
    As Chestermiller said, they most likely want you to pretend the acceleration is constant.
    (I'm curious how much larger the maximum acceleration would be than the average accel. in a real crash.)

    You need to be careful though with this equation though:
    x-x0 is the distance(s) that the passenger compartment(s) move(s) during the collision (which is not necessarily just the given 'crumple distance').

    Does the acceleration (that we want to find) differ in two frames moving at a constant speed relative to each other?
    The answer to this question is a hint as to how I would approach the problem.
  8. Jul 10, 2017 #7
    So that means the displacement would be 1.2 meters, because the passenger moves his .6 m and the other .6 as well, right?

    I'm trying to wrap my head around your question. Haha
  9. Jul 10, 2017 #8
    Can you appreciate the fact (qualitatively) that the smaller car should accelerate more?
    (It slows all the way down from 80km/hr and reverses its direction of motion, whereas the big car just slows down a bit from 80km/hr.)

    If you look at your equation,
    doesn't the fact that the accelerations should come out different for each car mean that (x-x0) should be different for each car?
    So how can we understand this difference in (x-x0) of each car?
    [Hints: does the contact point between the two cars move during the collision? Does this contribute to the displacement (x-x0)?]
  10. Jul 10, 2017 #9


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    I would put it differently. Since the duration of impact is not given, there is no easy way to calculate the average acceleration unless we assume it is constant. So just asking for "acceleration" is ok, but they ought to say assume it is constant.
    In fact, in a survivable head-on car crash it is more likely to be roughly constant than in many other collisions. That is because car extremities are designed to crumple at approximately constant load so as to minimise the maximum acceleration. That breaks down at unsurvivable speeds.
    Last edited: Jul 10, 2017
  11. Jul 10, 2017 #10


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    It might be conceptually simpler to calculate the duration of the impact first, using relative velocities and displacements.
  12. Jul 10, 2017 #11
    As you indicated, this is seems to be a pretty complicated problem. I have been trying to model this by putting a damper on the front of each of the cars to represent the (dissipating) crumpling region of each. I have started out by assuming that the velocity of the contact point between the two cars is equal to the final velocity. But, when I do this, and, requiring that, by Newton's third law, the contact force is the same for each car (at each instant of time), I find that the two crushing zones are not equal. The crushing region of the lighter car comes out larger than the crushing region of the heavier car. I'll keep working on this an keep you posted on progress.

  13. Jul 10, 2017 #12
    I made the same assumption, that 'the velocity of the contact point is the final velocity,' but I thought of it as, 'the contact point does not move in the center of mass frame.'

    In the CoM frame, if the contact point were stationary, the rate at which each car is crushed will be the same as the speed of the back of the car. Since the lighter car moves faster at all times, it will be crushed more at all times, and so we will not find the 'crumple distances' to be equal.

    Conclusion: The assumption that the contact point moves with the center of mass is inconsistent with the fact that they crumple the same amount.

    When I used that inconsistent assumption, my answer agreed with what the OP said was correct, so I now think his given answer isn't right. (I think @haruspex has the proper idea in post #10.)
  14. Jul 10, 2017 #13
    So how do I go about solving for time in this way? Kinematic equations still?
  15. Jul 10, 2017 #14
    The book says to assume the accelerations are constant. I'm not sure if that affects your answers in any way.
  16. Jul 10, 2017 #15


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    Yes. As I wrote, think in terms of the relative speeds and displacements. From the instant they make contact, how much closer do the drivers get, and at what average approach speed?
  17. Jul 10, 2017 #16
    In my judgment, the person who formulated this problem tremendously underestimated the complexity of what he was asking. I think he intended for the student to set the change in kinetic energy of each of the cars (as reckoned by an observer in the laboratory frame of reference) equal to the average force times the crumple distance. As pointed out by Haruspex, Hiero, and myself, this is an incorrect approach. It would be better to allow the two crumple distances to be different, and to specify the average of the two crumple distances. Then, at least, the analyses of Hiero and myself would be applicable.
  18. Jul 10, 2017 #17
    They each move 80 km/h towards each other, relative velocity of 160 km/h, and the displacement is 1.2 m (.6 m for each car). Is it as simple as 1.2 m divided by 160 km/h to get the time?
  19. Jul 10, 2017 #18
    It is very nearly that simple! That would be wrong though, because 160km/h is the relative speed at the initial moment. (At the final moment, for example, the relative speed is zero.) You'd want to use the average relative speed instead of the initial relative speed.
  20. Jul 10, 2017 #19
    Looking again, should I use the equation
    x-x0 = (1/2)(v0+v)t
  21. Jul 10, 2017 #20
    It depends what you mean by these variables. It can be correct, yes, if you have the correct meaning for the variables.
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