How do I catch someone falling from a short height without hurting or bruising them I am the backspot in the stunts for cheer and I keep catching our flyer but I am not absorbing her fall so she has bruises underneath her arms.  What is the physics of catching her without hurting her?
 A: Cheer coaches can give much better practical advice than we can,  but from a physics perspective the goal is to have a uniform de-acceleration over a long time and distance instead of an abrupt de-acceleration.   Once you receive them on the way down you want to slow them as smoothly as possible over the longest possible distance, spreading the force over the largest catch area. In cheer terms, the goal is "Catching high and absorbing well".
Bruises form under a wide variety of conditions, but a plausible rough criterion is that a bruise may form if the peak contact pressure is more than about a MPa and the total transferred energy per unit area is more than about $100\,\textrm{kJ/m}^2$.
If you throw a flyer up a distance $h$, when they come back down they will have a velocity $v=\sqrt{2gh}$. The constant de-acceleration $a$ needed to slow them down to a stop in a catching distance $s$ is
$$a=\frac{v^2}{2s}=\frac{h}{s}g$$ where $g\approx10\,\textrm{m/s}^2$ is the acceration due to gravity.
If you are catching the flyer under their arms, then the contact area might be about $300\,\textrm{cm}^2$. If a $50$ kg flyer is thrown up $3$ m, then their energy when they hit your arms is about 1500 J, so the energy transfer is only about $50\,\textrm{kJ/m}^2$.
If you slow them down smoothly over a $30$ cm vertical distance, then they are de-accelerating them at about $10$ g $\approx 100\,\textrm{m/s}^2$, which requires a force ($F=ma$) of $5000$ N, and the pressure over the catch contact area is only about $0.16$ MPa.  So a single smooth catch onto wide arms would not be expected to cause bruises.
If, however, you are throwing them higher (e.g. $4$ m), mostly slowing them down over a shorter distance e.g. $15$ cm, and most of the force is over a smaller area, e.g. $100\,\textrm{cm}^2$, then the energy transfer and pressure (e.g.$100\,\textrm{kJ/m}^2$, $1.3$ MPa) could certainly be high enough to cause a bruise.
