Real World Free-Fall Simulator Help Please

Jun 2018
Houston, TX
Here’s the scenario:

I have a 1.2kg dart that is dropped from a height of 50m and penetrates into the ground to a depth of 0.09m. I can calculate (with wind resistance), the velocity, the kinetic energy at impact, the momentum at impact. Using the 0.09m penetration depth, I can also calculate the amount of work done (after impact), the impact force, the rate of deceleration and the time required to decelerate.

I am designing a tool to simulate these results which looks a lot like a mechanical fence post driver with a mast and a falling weight (with an appendage that looks like the dart). So, for a given mass and height, I can calculate (with wind resistance), the velocity, the kinetic energy at impact, and the momentum at impact. My problem is that I don’t know which of these values I should be setting equal (if any) to get the same results from the tool as I do from the dart.

Can I assume the same depth of penetration as observed with the dart? If so, I can then calculate the required combination of mass and height to get the same work and impact force from the tool. Will this then give me the same depth of penetration?

Here are some values from my calculations:

Mass: 1.2kg
Initial height: 50m
Drag factor: 1.3
Time of free-fall: 3.3897 seconds
Final Veloclty: 27.8029 m/s
Final Momentum: 33.3635 kg*m/s
Work: 463.80 Joule
Impact Force: 553.33 kg*m/s^2

Mass: 40kg
Initial height: 1.203m
Drag factor: 2
Time of free-fall: 0.4975 seconds
Final Veloclty: 4.8156 m/s
Final Momentum: 192.6239 kg*m/s
Work*: 463.80 Joule
Impact Force*: 553.33 kg*m/s^2

NOTE* Using an assumed depth of penetration of 0.09m.

In this example, the work done and the impact force have been made equal. Does this mean that I will get the same depth of penetration?



PHF Helper
Jun 2010
Morristown, NJ USA
If I understand your question correctly, you are asking whether the simulating tool should result in the same momentum as the dart upon impact with the ground, or should it it have the same kinetic energy - is that right? My suggestion is to try to match the same KE, as the depth of penetration into the ground is controlled by the work required to dislocate the soil as the dart penetrates the ground. The net result is that if the tool has a mass that is X times larger than the dart, it should be dropped from a height that is about 1/X of the dart's (neglecting the effects of wind resistance).

By the way, you use the term "impact force" without defining what you mean. Are you assuing a constant force acting over the distance of penetration? If so then the calculation should be F*d = Work, or F = 463.8J/0.09m = 5153 N. But I don't think this is a very useful calculation, as the deceleration force most likely varies with the depth of penetration due to friction between dart and soil.
Apr 2017
I think the essence of this question is.. 'would a darts with the same kinetic energy (but vastly different velocities ) have the same penetration'

the answer has to be no .. the ground has to react differently to different penetration velocities .. but the difference in penetration may only be a few %...

this can only be resolved by experiment , and of course no patch of ground is the same as any other patch .