A change in momentum is called an impulse which is equal to force applied over time. It's not the change in momentum since the change in momentum to reach zero speed from a given speed is the same whether or not it happens over a longer or shorter period of time. You can use more force over less time or less force over more time.
But what this does change is deceleration. If you decelerate by quickly (i.e. achieve the same change in speed over a shorter amount of time), that requires more force and this higher force can cause exceed the structure's strength.
Springs in this cause are supposed to stretch out the deceleration over a longer period of time to decrease the force, but they also store the energy so won't be terribly helpful without a mechanism to allow the springs to dissipate that energy (it will send your robot bouncing back up only to fall again). That's why big machines have things like hydraulic dampers in addition to the springs. Obviously not of an acceptable on a drone.
I'd actually forget about springs. I'd use one of the following approaches instead:
landing gear that collapses and breaks in a controlled manner. This spreads out the deceleration over time and the act of breaking dissipates energy. Like crumple zones in a car. You could do this with more rigid materials but that might take too much engineering effort since they could end up being too strong. Easy collapsible materials that are also cheap (i.e. replaceable) to use include things such as foam and corrugated cardboard (but mind the direction of the corrugations since cardboard can be very rigid in one direction). Balsa is another material that is middle of the road in that it could be made quite strong but quite snappable but is more costly and would take longer to work with. You can also notch beams to adjust when and where the break occurs.
landing gear that has shear pins in it. The shear pins break to dissipate the energy and are easier to replace than all the landing gear like the first approach. Depending on the how the mechanism is built you could potentially have a sequence of shear pins (such as telescopic tube where the inner tube is held by a row of shear pins that are progressively weaker which would give a gradual extended collapse). Many weak shear pins that break in a sequence is better than one stronger one that breaks because it spreads the deceleration force out over time. You could make the first pin strongest since that carries the operating loads but if it breaks you know something is wrong and you want the others to break too to absorb the energy.
Just using plastic screws in your landing gear is a start. Depending on the weight you may have to use unconventional shear pin materials such as toothpicks that snap easily.
- collapsible landing gear held by a friction mechanism (like maybe a brass/nylon tipped setscrew or sandpaper clamp). The friction mechanism spreads the deceleration over time while dissipating energy. A telescopic tube, for example where the inner tube or rod is held by a tuned set screw or friction collar. Could also be combined with a weak shear pin at the very end just as a last ditch safety if even the friction clutch doesn't dissipate enough energy.
- If you want to get real fancy and make a hydraulic-type damper, what
these do is have a reservoir of oil and when you press the piston it
forces the oil through a small orifice into another chamber and this
dissipates the energy. A bit fancy but a DIY version would probably
involve a syringe with some liquid inside like baby oil (mineral oil)
or maybe even vaseline but that might be too thick. The main advantage of the vaseline is it is very viscous so is easy to control and keep in place. You have an orifice leading
to the outside that you seal off (perhaps even with a bit of tape if
it's vaseline since it isn't very fluid). If you press the plunger
hard enough the seal pops off and the liquid is forced through the
orifice into the outside rather than into another chamber.
