How can I calculate the compression of a ball?

Question

I was thinking about elastic collisions, and then I thought, What causes more compression in a ball when it's hit? Is it the velocity of the thing hitting the ball? Does the mass of the thing hitting the ball have something to do with the compression?

Here is an example:

Imagine a golf ball is hit by an object of $1 \text{kg}$, then we do the same experiment but with an object of $1000 \text{kg}$. The velocities are the same in the collision ($10$ $\text{m/s}$) but the mass is not. Will the ball have the same compression in both cases?

Answer 1

the constraint force F between the club and the ball is proportional to the momentum during the collision :

$$p=\int F\,dt\quad,\text{with}\\ p=2\,{\frac {m_{{2}}\,m_{{1}}{\it u_1}}{m_{{2}}+m_{{1}}}}$$

where

• $u_1~$ start velocity of the golf club
• $m_1~$ mass of the golf club
• $m_2~$ mass of the golf ball
• $p~$ linear momentum during the collision $~\left[\frac{\text{kg m}}{s^2}\right]$

thus if $~m_2\mapsto f\,m_2~$ you obtain that

$$\frac{p(m_2=f\,m_2)}{p}=\frac{p_f}{p}=\underbrace{{\frac {f \left( m_{{2}}+m_{{1}} \right) }{fm_{{2}}+m_{{1}}}}}_{\kappa}$$

hence the ball compression is $~\kappa~$ time grater then if $~\kappa~$ equal one ($f=1$)

$$p_f=\kappa\,p$$

Answer 2

The compression of an elastic ball is proportional to the applied force. At the moment of the collision the portion of the ball in contact with the object is forced to move immediately with its velocity, while the rest of it is accelerated by the force of compression.

Eventually, that acceleration results in an increase of the ball velocity that exceeds the object velocity, and the ball detaches.

During a given time $\Delta $ before they loose contact, the reaction force results in a decrease of the momentum of the object. As $\Delta p = m \Delta v$, a bigger $m$ results in a smaller reduction of velocity for the same decrease of momentum.

So, during that same time, the contact velocity keeps greater for a bigger mass, resulting in a bigger force and a bigger compression.