I am trying to understand a case of impact between a sphere and a half-space.
Suppose I have a particle resting on the half-space in the gravity. The penetration depth $\delta_{max}$ of resting particle can be obtained from force balance of $$F_g=F_{Hertz},$$ where $F_g=m_pg$, $F_{Hertz}=k\delta_{max}^{3/2}$, here $k=4/3E^{*}R^{1/2}$ which is constant, $m_p$ is mass of particle, $g$ is acceleration due to gravity, $E^{*}$ is effective elastic modulus, and $R$ is effective radius of half-space. Solving the force balance for penetration depth $\delta$ yields $$ \delta_{max}^{3/2}=\frac{m_pg}{k} \tag{1}$$
Now, let's assume that the particle had some initial normal velocity $v$. Therefore, kinetic energy $E_{kin}$ and try to calculate penetration depth. In this case, I am not sure if we can make a balance of forces. I found a more common approach, which is to transform forces into work by integration $W= \int F \ d \delta $. If the contact is fully elastic and no energy is lost, the whole kinetic energy of impact needs to be counteracted by the work done by contact force. Therfore we can formulate energy balance $$ E_{kin} = \int_0^{\delta_{max}} F \ d \delta $$ for now ignore gravity and solve only for contact forces so $F=F_{Hertz}$ thus $$ E_{kin} = \int_0^{\delta_{max}} k\delta^{3/2} \ d \delta $$ this yields $$ E_{kin}= [\frac{2}{5} k\delta^{5/2} + C]_0^{\delta_{max}}$$ integration constant $C$ vanishes and we get solution
$$ \delta_{max}^{5/2}= \frac{E_{kin}}{k} \frac{5}{2} $$
- my problem starts when we assume that gravity plays a role, i.e., in the case when normal impact velocity is small; thus, $E_{kin}$ is small, and the contact is dominated by gravitational force. Using the same approach as in the second case and assuming that gravity acts in the opposite direction from contact force. I get an energy balance $$ E_{kin}=\int_0^{\delta_{max}} k\delta^{3/2} - m_pg \ d \delta $$ the integration result is $$ E_{kin}= [\frac{2}{5} k\delta^{5/2}- m_pg \delta + C]_0^{\delta_{max}}$$ integration constant $C$ vanishes and we get solution $$ E_{kin}= \frac{2}{5} k\delta_{max}^{5/2}- m_pg \delta_{max}$$ I found it strange that if we assume that $\lim_{E_{kin} \to 0}$ and rearrange above equation we get $$\frac{2}{5} k\delta_{max}^{5/2}= m_pg \delta_{max} $$ simplifying for $\delta_{max}$ gives $$ \delta_{max}^{3/2}= \frac{m_pg}{k} \frac{5}{2} \tag{2}$$ Now, when I compare the results from equations $1$ and $2$, which I expect to be identical, I find that I am off by a factor of 5/2.
This discrepancy is puzzling, and I would appreciate any insights into what I might be missing.
Edit: I realized that I should refine my question. The inconsistency arises from the integration process. If I choose not to integrate force but instead convert kinetic energy to force, I assume that when $E_{kin}$ is zero, the converted force $\frac{d \ E_{kin}}{d \delta}$ would also approach zero, thus preventing the inconsistency. My question is: why can't I integrate the force and create an energy balance for the system?
