Ask your DM if they rule that the movement or positioning of falling prone has to require appreciable effort or if it has to require any movement at all.
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Speed only determines how far you can move.

**Speed**, PHB 181:

> Every character and monster has a speed, which is the distance in feet
> that the character or monster can walk in 1 round. This number assumes
> short bursts of energetic movement in the midst of a life threatening
> situation. The following rules determine how far a character or
> monster can move in a minute, an hour, or a day.

But this condition is explicit in that you can't move, **at all**, logically you can not do any of the Movement and Positioning under which Being Prone is listed (PHB 190).

There is a distinction in terminology that is made regarding [Conditions][1], PHB 290-292.

Petrification lists the distinct states, Petrified, PHB 291: 

> The creature is incapacitated (see the condition), can’t move or
> speak, and is unaware of its surroundings.

Distinctions are:
 
1. Incapacitated.
2. Can’t move.
3. Can't speak.
4. Is Unaware (of surroundings).

Being grappled further indicates that a bonus to a creatures speed would resolve the issue of having 0 speed, it could, therefore, move **Grappled**, PHB 290:

> A grappled creature’s speed becomes 0, and it can’t benefit from any
> bonus to its speed.

Having 0 speed and not being able to move is not the same. The different states of being petrified showcase the difference in the agency that is left to you.

**Is there a case for you being able to drop prone without being able to move?**

Yes, there is!


There are two philosophies, according to this one you can drop your weapon and not count that weapon as an object and therefore make no object interaction because 

> letting go of something requires no appreciable effort. But picking it
> up does.

As argued in [this thread][2], the quote refers to this [sage advice][3].

(The other is that weapons are objects)

So if letting go of an object costs you no object interaction because there is no appreciable effort required, then you could argue that there also is no appreciable effort required from moving from standing into dropping down - just relax your muscles.

You may also ask - if you follow this philosophy - if you are subject to fall damage.


  [1]: https://roll20.net/compendium/dnd5e/Conditions#content
  [2]: https://rpg.stackexchange.com/questions/70237/is-dropping-a-weapon-free
  [3]: https://www.sageadvice.eu/2017/03/29/what-are-the-rules-on-dropping-weapons/