Can you use mana from a Dark Ritual to cast two spells in the same turn?

Dark Ritual adds three black mana to your mana pool when it resolves. Mana in your pool persists until you use it or until a phase or step ends, at which point any unspent mana empties from your pool (CR 500.4). This means you can absolutely use that mana to cast two spells in the same turn — as long as you spend it before the phase ends.

The key constraint is timing. You can only cast sorcery-speed spells during your main phase when the stack is empty (CR 307.1). So if you want to cast two sorceries with Dark Ritual's mana, you cast the first sorcery, let it resolve, and then cast the second — all in the same main phase. Instant-speed spells have more flexibility but the mana still must be spent before the phase ends.

It's also worth noting that mana pools don't carry over between phases or steps (CR 500.4). If your first spell resolves and you move to a new phase without spending the remaining mana, that mana is gone. Plan your sequencing carefully.

Concrete example: You cast Dark Ritual, it resolves, and you now have three black mana in your pool. You spend two of it to cast Hypnotic Specter (a 2B creature). It resolves. You then spend the remaining one black mana to cast Duress (a 1B sorcery... wait, Duress costs 1B — you'd need more). Instead, you cast Dark Ritual, netting {B}{B}{B}, cast Hypnotic Specter for {2}{B}, then use the last {B} to cast a Thoughtseize for {B}. Both spells are cast in the same main phase using mana from the single Dark Ritual.

So yes, Dark Ritual's mana can fuel two spells in one turn as long as their combined cost fits within the three mana produced and you cast them during the appropriate timing windows before the phase ends.