Potrzebujemy przestrzeni do ćwiczeń.

Because Polish often drops subject pronouns when they are already clear from the verb form.

Here, potrzebujemy ends in -my, which tells you the subject is we.

So:

• Potrzebujemy = we need

You can add my if you want emphasis or contrast:

• My potrzebujemy przestrzeni do ćwiczeń, a oni nie.
  We need space for exercise, but they don’t.

Potrzebujemy is the 1st person plural present tense form of potrzebować.

• potrzebować = to need
• potrzebujemy = we need

In this sentence it simply means a current or general need: we need.

Polish does not use a separate helping verb like English sometimes does. One verb form is enough.

Because potrzebować normally takes the genitive case.

The dictionary form is:

• przestrzeń = space

But after potrzebować, it changes to the genitive:

• potrzebować przestrzeni = to need space

So:

• Potrzebujemy przestrzeni = We need space

This is one of the key patterns to remember:

• potrzebować + genitive

That is simply the standard grammar pattern of this verb in Polish.

Many Polish verbs require a particular case, and potrzebować is one of the verbs that typically governs the genitive.

Examples:

• Potrzebuję czasu. = I need time.
• Potrzebujesz pomocy. = You need help.
• Potrzebujemy przestrzeni. = We need space.

For an English speaker, this can feel unusual, because English does not mark nouns this way after need. In Polish, though, this is completely normal.

Because Polish does not have articles like English.

So przestrzeni can mean:

• space
• some space
• the space

The exact meaning depends on context.

That means English has to choose an article when translating, but Polish does not need one.

Here do does not mean movement toward something. In this sentence, do + genitive shows purpose or intended use.

So:

• przestrzeń do ćwiczeń = space for exercise / for practicing

This is a very common Polish pattern:

• buty do biegania = shoes for running
• maszyna do szycia = sewing machine
• pokój do pracy = room for work

So do ćwiczeń is best understood as for exercise, for training, or for practice, depending on context.

Because do requires the genitive, and here the noun is in the plural genitive.

Base form:

• ćwiczenie = exercise / drill / practice task

Plural:

• ćwiczenia = exercises

Genitive plural:

• ćwiczeń

So:

• do ćwiczeń = for exercises

In natural English, though, that will often be translated more smoothly as:

• for exercise
• for training
• for practice

depending on the situation.

Because Polish often uses a plural noun in places where English prefers a singular or an uncountable word.

Literally, do ćwiczeń is for exercises, but idiomatically it may mean:

• for exercise
• for working out
• for practice
• for drills

So the Polish plural does not always have to sound plural in English.

No. Ćwiczenie / ćwiczenia can refer to different kinds of practice, depending on context.

It can mean:

• physical exercise
• training exercises
• drills
• practice tasks
• rehearsal-type exercises

So przestrzeń do ćwiczeń could mean space for:

• working out
• training
• rehearsing
• practicing movements or tasks

The exact meaning depends on the broader context.

Yes, and that is very natural too.

The difference is mainly one of nuance:

• miejsce = place, spot, location
• przestrzeń = space, room, area

So:

• Potrzebujemy miejsca do ćwiczeń = We need a place to exercise.
• Potrzebujemy przestrzeni do ćwiczeń = We need space to exercise / room for exercise.

Przestrzeń emphasizes available room or area, while miejsce emphasizes a place more generally.

No, Polish word order is fairly flexible.

The neutral, natural version is:

• Potrzebujemy przestrzeni do ćwiczeń.

But you could also move things around for emphasis:

• Do ćwiczeń potrzebujemy przestrzeni.
  This puts more focus on for exercise.

• Przestrzeni do ćwiczeń potrzebujemy.
  This sounds more marked and emphatic.

Because Polish uses case endings, the grammar is still clear even when the order changes. The main difference is usually emphasis, not basic meaning.