I know the coefficient of restitution depends on the velocity before and after the impact of a bouncing ball. How does the mass affect it. Can't figure it out.
v is the scalar velocity of the object after impact
u is the scalar velocity of the object before impact
and the H and h are related to the heights. How can I relate it to the mass of the object?
The masses affect the magnitude of impulse in a collision.
Consider two bodies of mass $m_1$ and $m_2$ about to collide with relative velocity $v_{\rm rel} = v_2 - v_1$.
Considering the coefficient of restitution $\epsilon$, which is an empirical value is used to find the relative velocity after the collision
$$v_{\rm rel}' = - \epsilon\; v_{\rm rel} \tag{1}$$
the required impulse $J$ to follow this rule is simply
$$ J = (1+\epsilon) m_{\star} v_{\rm rel} \tag{2}$$
where $m_{\star}$ is the reduced mass of the system and it is defined as $$ m_\star = \frac{1}{ \tfrac{1}{m_1} + \tfrac{1}{m_2} }$$
The final velocities are found by the application of an equal and opposite impulse $J$ to each body
$$ \begin{aligned} v_1' &= v_1 + \tfrac{1}{m_1} J & v_2' &= v_2 - \tfrac{1}{m_2} J \end{aligned} \tag{3}$$
If you plug (2) into (3) you will find that $J$ solves the equation (1) exactly.
In summary, the coefficient of restitution $\epsilon$ is a simplification that is empirically derived from experiments relatives the relative motion before and the collision. Given a COR value and the conditions of a collision the impulse magnitude) is estimated from (2).
Additionally, if the collision is estimated to occur over a finite time $\Delta t>0$ then the impulse $J$ can give us an estimate of the peak and average contact force
$$ \begin{aligned} F_{\rm ave} & = \frac{J}{\Delta t} & F_{\rm max} & = \frac{\pi J}{2 \Delta t} \end{aligned} $$
This is derived from the estimated force-over-time curve to be that of half a cosine.
The coefficient of restitution of the ball can be expressed as the ratio of momenta before and after impact. Since the mass of the ball does not change, it cancels out leaving only the ratio of velocities.
The coefficient of restitution is defined as the ratio of relative velocities of colliding bodies before collision with relative velocity of colliding bodies after collision.
As these velocities are not dependent on the masses of the colliding bodies, the coefficient of restitution is also independent of the bodies' masses.
