How to compare the force of different impacts on a punch bag? I can read acceleration, speed, and displacement of an object over time. What parameters best describe force and speed of an impact with this object?
Note: I don't need absolute force values but a way to compare force and speed of different impacts. 
Specifically, the object is a mounted heavy bag and the acceleration is calculated on a certain point on the surface of the bag.
I am aware it's an ill-posed problem, but I am looking for a "good enough" solution.
Thank you in advance.
 A: A "good enough" solution would be to use the punch bag as a ballistic pendulum.
The easiest measure of the strength of the punch is probably its kinetic energy. The punch transfers kinetic energy to the bag, which recoils by swinging through an angle. This raises its centre of mass (CM) through height $H$ metres such that the potential energy gained equals the kinetic energy $K$ imparted by the punch :
$K=MgH=MgL(1-\cos\theta)=2MgL\sin^2\frac{\theta}{2}$
where $M$ in kg is the mass of the bag, $g=10m/s^2$ is the acceleration due to gravity, $L$ is the distance in metres of its CM below the support point, and $\theta$ is the maximum angle through which the CM swings. Note that $\theta$ is the angle from the vertical of a line from the support point to the CM of the bag, not the angle of the support rope.
If you are only concerned about the relative impact of two different punches, you can calculate the ratio $(\sin(\frac12\theta_1)/\sin(\frac12\theta_2))^2$.
The recoil could be recorded from the side on a smart 'phone, and the maximum angle $\theta$ measured on the appropriate frame against a plumbline in the background.
Some drawbacks of this method are :  


*

*The bag is often very heavy and does not swing very far even for the heaviest punch. If this is the case it is difficult to distinguish between punches. Ideally the strongest punch should swing the bag through $60^{\circ}$ or more. Consider reducing the mass $M$ of the bag and/or the distance $L$ of the centre of mass from the support.  

*The swing might not be exactly perpendicular to the camera. This could be checked by making a 2nd video from the front at the same time, at $90^{\circ}$ to the 1st recording.    

*If the punch is not delivered at the CM the bag will spin about a vertical axis. This should be avoided, because it takes kinetic energy away from the swing, reducing the measurement of impact. Use a coloured spot on the bag to identify spin on the video. If the spot moves more than $\frac14$ turn before the swing reaches maximum, take the punch again.  

*Another confounding motion reducing swing angle $\theta$ is rotation about a horizontal axis. The axis of the punch bag should always point to the support point throughout the swing.  

*Not all of the kinetic energy in the punch causes the bag to swing. The bag is designed to absorb energy by deforming. However, for a given punch bag approximately the same fraction of kinetic energy is lost for all punches. You should be cautious about comparing $K$ measured with punch bags which differ in size or design.


Kinetic energy is not the same as force, which varies during the punch or kick. The average force $F=Ma$ imparted to the bag can be calculated from its acceleration $a$. The accelerations of the top and bottom of the bag immediately after the punch can be measured with embedded accelerometers and the average used as $a$. 
Both methods are discussed in How to quantify martial arts strikes? The use of accelerometers is described eg in Measuring the force of punches and kicks among combat sports athletes using a modified punch bag with embedded accelerometer. 
