My group had a question that I figured I would ask here. If I have someone who is affected by a movement impairing ability making their movement zero and they are knocked prone, are they essentially stuck because moving from prone requires them to spend half of their movement speed? Or is it half of zero is zero, so he can still stand up. I look at movement speed like a container, and when the container is empty you cant take half of nothing. 

The usual interpretation is that if your speed is reduced to zero - you can't move.

Since standing up requires you to move, but a zero speed prevents movement. You can't stand up.

Standing up requires you to expend 1/2 of your movement but my take is that this assumes that you have a current speed greater than zero so that you actually have some movement to spend.

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One lockdown tactic used by PCs is to grapple a creature and then knock it prone. Since its speed is zero due to being grappled it is unable to stand up.

Quote from Lyxen >>

If it's sentinel, the speed drops to zero, and standing up from prone is actually based on the speed of the character. And it's true that half of 0 is still 0, so, RAW, I don't think that RAW provides a solution, unless you factor in this:

If you have more than one speed, such as your walking speed and a flying speed, you can switch back and forth between your speeds during your move. Whenever you switch, subtract the distance you've already moved from the new speed. The result determines how much farther you can move. If the result is 0 or less, you can't use the new speed during the current move.

 

Seeing the above, your speed becomes 0, you want to change to a new "movement mode" by standing up, but you can't because the speed is 0 or less.

This is in line with the logic that says that if you cant move, you certainly not be standing up for free. 

Finally, standing up for free would be ridiculous because you can also drop prone for free, so basically you could do an infinite number of burpees in 0 time... :p

 You aren't switching between different movement types by standing up but logic still dictates that zero movement means you can't move at all. Unfortunately we don't get much help by the PHB since the section "Climbing, Swimming and Crawling" on page 182 only gives us rules for climbing and swimming.

I always considered 'half' movement to be half of your maximum speed, not of what you have left. In that case it would not be half of 0, but half of (presumably) 30.

Edit:

From basic rules:

Standing up takes more effort; doing so costs an amount of movement equal to half your speed. For example, if your speed is 30 feet, you must spend 15 feet of movement to stand up. You can't stand up if you don't have enough movement left or if your speed is 0.

Quote from account256 >>

I always considered 'half' movement to be half of your maximum speed, not of what you have left. In that case it would not be half of 0, but half of (presumably) 30.

Edit:

From basic rules:

Standing up takes more effort; doing so costs an amount of movement equal to half your speed. For example, if your speed is 30 feet, you must spend 15 feet of movement to stand up. You can't stand up if you don't have enough movement left or if your speed is 0.

 Yeah, just this. It is a somewhat well known battle strategy to grapple and shove (pin) a creature to trap them prone.