The initial condition of the model system.
The total number of collisions N as a function of the mass ratio . The values and are the minimum values of to have N = 1 and N = 2, respectively. As , which diverges to infinity.
(a) The collision points at , where and N = 3. The trajectory of the block x(t) consists of line segments that connect every collision point . The lower bound f(t) passes every collision point and the upper bound g(t) is tangent to every line segment . The continuum representation of the trajectory is the arithmetic average of g(t) and f(t). (b) The same as (a) except that , and N = 5.
The values of , and for , and . The values at and can be obtained by using identities such as and .
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