For the ball and paper sheet experiment, the anomalous height may be the result of a reflected wave from the initial impact.

This simulation provides an idealized model to help isolate this as the main effect. Instead of paper sheets and a ball, we idealize the system as a 1D chain of particles with damped harmonic contacts, impacted by a single free particle under gravity.

The following parameters can be tuned to investigate this effect:

• $\hat{m} = \frac{m_b}{m_c}$ = ratio of ball to chain mass
• $\hat{k} = \frac{k_b}{k_c}$ = spring constant of the ball
• $\hat{d} = \frac{d_b}{d_c}$ diameter ratio of ball to chain
• $N$ = number of particles (already dimensionless)
• $\hat{\gamma} = \frac{\gamma}{\sqrt{k_c, m_c}}$
  • When $\hat{\gamma} >1$ means overdamped springs.
• $\hat{v} = \frac{v_0}{d\sqrt{\frac{k_c}{m_c}}}$ =
  • When $\hat{v}>1$, means that the kinetic energy from impact is greater than potential energy of the spring.

$$ \hat{v} = \frac{v_0}{d\sqrt{\frac{k_c}{m_c}}} \rightarrow \hat{v}^2 = \frac{v_0^2,m}{d^2k} $$

• $\hat{g} = \frac{g}{d_c^2\frac{k_c}{m_c}}$
  • When $\hat{g}>1$, means that the gravity is stronger than force to full compress the chain spring

$$ \hat{g} = \frac{g}{d^2\frac{k_c}{m_c}} = \frac{gm}{d_ck_c} $$

• $e$ = coefficient of restitution (dimensionless)
• $\tau$ = contact time (time -TBD)

Results with Appropriate Physical Parameters

With parameters that match those in the lab:

• $\hat{m}= 20$; the mass of the ball is much more than the mass of the paper-cells.
• $\hat{\gamma} = [.0006, 0023]$; damping should be included, but is not a dominating factor
• $\hat{v}= .1$; impact kinetic energy is 10% of the potential energy of the springs.
• $\hat{g} = 0$; gravity is not a significant factor of the experiment

We get the following curves:

The actual visualization of the simulations are shown in the following video.

• White circles
  Default state for all particles: no special event has occurred yet.

• Yellow circle (particle 2, just below the ball)
  Turns yellow when the reflected wave is predicted to arrive back at the top of the stack (based on the measured travel time to the bottom). This highlights the moment when energy carried by the wave reaches the upper contact again.

• Red circle (bottom particle)
  Turns red once the wave first reaches and compresses the bottom contact. This marks the moment the initial impact pulse has traversed the entire stack.

• Cyan circle (ball)
  The ball turns cyan after it has lost contact with the stack (gap larger than one diameter plus a small tolerance). This visually marks the onset of rebound / free flight.

plot.mp4

Pressure Effect

Decreasing pressure effectively decreases spring constant of the paper-air cell. Because effective damping and spring constant are coupled

$$ \hat{\gamma} = \frac{\gamma}{\sqrt{k_c, m_c}} $$

decreasing $k_c$ would also cause a decrease in $\hat{\gamma}$. Plotting this explicit, we see,

this causes a shift in peak to lower $N$ as well as decreasing overall restitution.