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In an environment without gravity, there is an unfixed, motionless, uniform spring with non-negligible mass m, length L, and stiffness k suspending and a rigid cube with mass M moving with velocity v toward the spring. Both of the spring and the cube can move freely, and there is no other object and any external force in this system. As shown in the figure, the moving direction of the cube and the axis of the spring are on the same line. Assuming that the total mechanical energy is conserved, how to estimate the kinetic energy loss of the cube after the collision ?

enter image description here

The only strategy I know is to write down the equations of the spring's properties, which are PDEs, and solve it to make the velocity and mass density at any point on the spring explicit at any moment, so that I can calculate the total momentum of the spring when they are about to separate. Then I can estimate the kinetic energy change of the cube.

My approach:

I define two independent variables, t and s, which is the time variable and position variable of the frame of reference of a motionless observer.
Then I define several dependent variables, F(s,t), the internal force of the spring, a(s,t), the acceleration of an spring element at s, v(s,t), the velocity of an spring element at s, u(s,t), the linear density of an spring element at s.


What these definitions imply is that spring elements are stationary in space, whose idea is similar to the Eulerian specification in fluid field. On the other hand, "spring segment" in the following means a moving spring segment which is a certain actual part of spring, just like a specific flow parcel in the Lagrangian specification in fluid field.


Then I define two temporary differential elements used in derivation, dl, the original length of the spring segment in ds and dx, the difference of dl and ds.
Finally, I define a constant k1, which is independent of the original length of any spring segment and a constant D1, the original linear density of the spring.

Then I can write down several relationship directly.

enter image description here

Next, using the relationship between spring elements and spring segments, similar to the material derivative in fluid field, and the relationship above, I can write down two PDEs with Hooke's law and Newton's second law.

enter image description here

I'm still studying how to solve it, but at the same time, I'm wondering whether my approach is correct, and if there is a simpler solution.

p.s. I'm not a native English speaker, sorry for any improper use of words.

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  • $\begingroup$ Possible duplicate of Elastic collision and spring $\endgroup$ Commented Jun 9, 2018 at 14:45
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    $\begingroup$ You ought to show your attempt to solve the problem and explain what difficulty you are having. An ideal spring conserves energy. But there will be maximum loss of energy if the spring and block stick together. More information is required about the nature of the collision. $\endgroup$ Commented Jun 9, 2018 at 14:56
  • $\begingroup$ Unclear: Is the left end of the spring against an infinite mass, or is it just sitting there in space? $\endgroup$
    – Hot Licks
    Commented Jun 10, 2018 at 11:13
  • $\begingroup$ The left end of the spring can move freely, but it's motionless at the beginning. $\endgroup$
    – benexter
    Commented Jun 10, 2018 at 11:53
  • $\begingroup$ +1 I don't follow your analysis, but dividing the spring into segments is correct. It is a difficult problem, there may be no simple solution. Try solving a heavy spring colliding with a fixed wall. Model the heavy spring of mass $M$ as a series of $N$ identical masses $m=M/N$ connected by massless springs, then let $N \to \infty$. See scribd.com/document/142152314/Multi-Mass-Spring-Modeling . Also mathrec.org/old/2001dec/solutions.html and Effective mass in Spring-with-mass/mass system from which the previous 2 links came. $\endgroup$ Commented Jun 10, 2018 at 21:01

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