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Pegar vs. apanhar

The verbs apanhar and pegar can both translate to to take, to catch, or to pick up, but their usage varies significantly between Brazil and Portugal. In general, Brazilian Portuguese relies heavily on pegar for almost all contexts, whereas European Portuguese makes a distinct separation between specific types of actions. Most importantly, apanhar has a meaning of physical punishment in Brazil that can lead to awkward misunderstandings.

Pegar

A1
In Brazil, pegar is the universal verb acting as a synonym for to take, to catch, or to fetch, applying to everything from transport to illnesses. In Portugal, pegar is more specific; it usually refers to the physical act of holding, touching, or grabbing something with your hands, often accompanied by the preposition em.
Eu preciso pegar o ônibus para ir ao trabalho.
(I need to take the bus to go to work.)
Você pode pegar o livro na estante?
(Can you get the book from the shelf?)
Não gosto de pegar em dinheiro sujo.
(I do not like to touch dirty money.)
Vamos nos pegar às oito horas.
(We will pick each other up at eight o'clock.)
Ele vai pegar uma gripe se sair sem casaco.
(He will catch a flu if he goes out without a coat.)

Apanhar

A2
In Portugal, apanhar is the standard verb used when catching public transport, catching an illness, or picking something up from the floor. However, in Brazil, apanhar almost exclusively means to get beaten or to receive physical punishment. Using it to mean to take the bus in Brazil sounds very strange or archaic to locals, although they will understand the context.
Nós vamos apanhar o comboio para Lisboa.
(We are going to catch the train to Lisbon.)
Se você fizer isso de novo, vai apanhar.
(If you do that again, you will get beaten.)
Vou apanhar a caneta que caiu no chão.
(I am going to pick up the pen that fell on the floor.)
Coitado, ele acabou de apanhar na rua.
(Poor guy, he just got beaten up on the street.)
Espero não apanhar chuva hoje.
(I hope I do not get caught in the rain today.)

Summary

To sound natural and avoid confusion: use pegar for almost everything in Brazil (transport, objects, grabbing). In Portugal, distinguish between them: use apanhar for catching transport or picking things up, and pegar when you are physically holding or handling an object. Be very careful with apanhar in Brazil, as it implies violence.