While reading your question I get the feeling there are some misconceptions here, so let me start by clarifying a few things:
Paper cups are not solely made of paper but have a plastic or wax lining. It's to prevent the cup from failing apart when it gets wet,
but it also prevents or slows down (bio)degradation.
Plastic degrades into increasingly smaller pieces of plastic, primarily under the influence of sunlight and heat.
Paper on the other hand biodegrades when micro-organisms, fungi and worms turn the paper into CO2, water and compost.
Although scientists recently found bacteria that can 'eat' a specific type of plastic like PET very slowly (see this question), it's still uncommon to refer to degradation of traditional plastics like PP, LDPE, PET as biodegradation.
The exception is if you are referring to plastics that were specifically designed to be biodegradable.
Not all paper biodegrades quickly. Laminated paper, waxed paper, large quantities of paper, or paper put in dry or cold conditions will biodegrade much slower. I'll explain more below.
How fast something biodegrades depends on many variables, but an important aspect is the availability of micro-organisms. This in turn is influenced greatly by
the amount of oxygen and water that is present, and the ambient temperature (among others). Paper, or any other compostable material for that matter, will degrade more quickly if you do one of the following:
- Make sure the materials stay moist (not dry or soaked)!
- Keep temperatures around 40-60 degrees C
- Bury it under other degrading or already-degraded materials, so more surface of the material is exposed to micro-organisms (and cut thick material into pieces)
- Aerate regularly by turning the materials
- Keep a carbon-nitrogen balance of materials of around 25:1
Most plastics on the other hand will degrade most quickly if you expose it to sunlight (UV radiation and heat).
For your experiment the choice of environment will affect the outcome. Any situation you choose will most likely be more favorable for either plastic or paper.
I think the fairest comparison is when you try to mimic what generally happens with litter; just leave it exposed somewhere between or on top of organic materials like grass or dirt.
You could place it on some stones similar to litter in a gutter or on the pavement, but then you'd probably have to wait much longer for anything to happen.
I would suggest creating some sort of small enclosing on your land with chicken wire or perhaps something with a finer mesh. Basically anything to keep the materials you put into it from being blown away,
but still leave them exposed to the elements. Then just leave the materials alone. For pure paper without any finish or lining you want to check weekly what's happening. For plastics or paper cups it will most likely take quite some time
before anything happens, so at first I'd check that monthly.
Even though it would be most interesting to compare degradation of a plastic cup with a paper cup, I do not recommend using any traditional plastic or a plastic-lined paper cup.
When the plastic degrades it will contaminate the soil with the toxic additives that are present in the plastic. Also the tiny pieces of (micro)plastics are very hard or impossible to remove.
My own experiences
For what it's worth, I regularly add paper napkins or cardboard to my worm bin to keep a proper carbon-nitrogen balance in the bin. I've had very mixed results depending on the type and thickness of the paper or cardboard
and on the time of year. In summertime a small piece of paper napkin buried in the bin will be gone in 2-3 weeks. Multiple pieces of thick cardboard placed along the inside edges of the bin in late autumn (as extra insulation)
remained fairly intact for up to a year. Also a small piece of compostable plastic (most likely PLA) remained fully intact after being in the bin for more than a year. I recently bought something that came in a certified 'home-compostable' plastic wrapper so I started a new experiment and added a few pieces to my worm bin, but I have yet to check how that is going.