# Establishing impact force from deceleration data

I am trying to establish an impact force. The impact came from a spear (wooden with a stone tip) hitting a soft (meat) target. The acceleration was measured by a GCDC X400-2 accelerometer.

Due to the spear impacting a non-consistent fluid/solid target, the deceleration was not regular. This is the reason I do not know how to apply the F=ma formula that is clearly the one need here.

I need to establish the amount of force with which the spear impacted the target and the amount of energy that was dissipated throughout the impacting into the target.

I was thinking about using the energy in momentum formula, but this goes into relativism, which is beyond me.

Below is the acceleration data for the impact. It is inverted, due to accelerometer mounting position:

Time m/s^2 31.721039 -5.568076923 31.726013 -12.8724359 31.731018 -23.52961538 31.736023 -22.93089744 31.741028 -15.62653846 31.746002 -2.394871795 31.751007 4.67 31.756012 -3.472564103 31.761017 -9.878846154 31.766022 -14.54884615 31.770996 -21.97294872 31.776001 -21.61371795 31.781006 -12.33358974 31.786011 -7.005 31.791016 -9.998589744 31.79599 -8.920897436 31.800995 0.838205128 31.806 18.56025641 31.811005 81.30589744 31.81601 173.3887179 31.820984 153.9902564 31.825989 49.87320513 31.830994 7.364230769 31.835999 -38.19820513 31.840973 17.06346154 31.845978 138.0044872 31.850983 75.13910256 31.855988 -12.21384615 31.860993 -3.771923077 31.865967 123.455641 31.870972 229.1892308 31.875977 149.44 31.880982 26.58307692 31.885956 -15.32717949 31.890961 0 31.895966 -17.00358974 31.900971 -20.65576923 31.905976 0.478974359 31.91095 -3.233076923 31.915955 -0.538846154 31.92096 -5.268717949 31.925782 -12.51320513 31.930786 -10.29794872 31.935791 -4.610128205 31.940949 2.45474359 31.945954 -3.652179487 31.950959 -10.59730769  The average velocity of this spear in flight was 13.89 m/s. This was calculated as the average velocity from the time of release to the time of impact. This throwing distance was 5m and the flight time was 0.36s. There was no deceleration noticed during this flight time.

For reference, this is for a PhD in Archaeology and hence I am not the best at physics in the world!! If there is any more information that you need, please contact me and I will be happy to provide it (along with larger data samples if that would help. I cropped the data to keep this post smaller.)

Cheers!

• What archaeological question are you trying to answer? Impact force might not be the best measure of what you are trying to quantify. The amount of energy which was dissipated is probably a better measure; this is the kinetic energy of the spear, $K=\frac12 mv^2$ where $v=13.89m/s$. Sep 21, 2017 at 19:03
• The best you can do is average force over a time step. Actual peak force requires a lot more detailed modeling to get to. Sep 2, 2021 at 15:46
• What is the mass of the target and the mass of the spear. Sep 2, 2021 at 20:57

What seems interesting is the maximal force that the target undergoes. Using Newton's third law, you only have to know the mass of the spear, and its acceleration.

The force during the deceleration is not constant, but the instantaneous force is the acceleration value multiplied by the mass. The acceleration peaks at $$a=229\mathrm{~m/s^2}$$. Multiply this by the mass.

You could take the mean acceleration over the time interval 3.80 .. 3.88 s, which is about $$\overline a = 72~\mathrm{m/s^2}$$. The velocity change in this interval is $$\Delta v = \overline a \cdot \Delta t = 5.8~\mathrm{m/s}$$, which is quite a bit lower than the flight speed that you state (13.9 m/s), so either the spear went straight through the target and continued flying, or your accelerometer was broken.

For the `performance' of a spear, momentum transfer is probably a better metric than force, peak force, or kinetic energy. Momentum transfer can be expressed in various ways: $$\Delta p = m\Delta v = F \Delta t = m\overline a \Delta t$$.

Integrate the acceleration data to get change in speed over change in time I see a $$\Delta v = (4.99)-(-0.92) = 5.91 \;{\rm m\, s^{-1}}$$ over a time of $$\Delta t = (31.886)-(31.801) = 0.085\;{\rm s}$$

Take the translating mass $$m_{\rm target}$$ of the target where the accelerometer is mounted and convert the change in speed into an impulse (unit of momentum change)

$$J = m_{\rm target} \Delta v$$

The change in momentum relates to the average force applied as

$$F_{\rm average} = \frac{J}{\Delta t}$$

Now to find the peak force, we see there are 3 peaks of similar magnitude, and we can estimate the impulse of each peak $$J_i$$ as well as the time each peak lasted $$\Delta t_i$$

Fitting a basic cosine wave on the bounce data you will find the peak force for each peak as

$$F_i = \frac{\pi}{2} \frac{J_i}{\Delta t_i}$$

Using these formulas, and an example target mass of $$70 {\rm kg}$$ I get the following values for peak force 