In German, there are common verbs that you will hear in relation to pain and injury, but also certain colloquial expressions. 

Ist jemand verletzt?

Is anyone injured?

Caption 35, Deutsch für Anfänger: #10 Notfall

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Wer städtisches Recht verletzt hatte, flüchtete zum Domnapf.

Anyone who had violated municipal laws fled to the Domnapf.

Caption 40, Jenny zeigt uns: den Speyerer Dom

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When the focus is really on the pain, the best verb to use is weh tun (the spelling wehtun is also accepted).

Es tut weh. -Der Arzt schaut sich das gleich an.

It hurts. -The doctor will examine it in just a moment.

Caption 13, Nicos Weg: Beim Arzt

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Sie tun mir weh!

You're hurting me!

Caption 2, Die Pfefferkörner: Abi um jeden Preis

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With the verb schaden, the emphasis is really on damage. Below you can also see that the German counterpart of the common English phrase "It can't hurt" utilizes schaden.

Doch ist für Experten wie Schmolz noch völlig offen, ob die Papageien den heimischen Vogelarten schaden.

But indeed, for experts, like Schmolz, it still remains an open question, whether or not the parrots are damaging the native bird species.

Captions 19-20, Freilebende Papageien: Überwintern in Wiesbaden

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Ein wenig neugierig zu sein kann manchmal gar nicht schaden.

Sometimes, indeed, it can't hurt to be a little curious.

Caption 88, Märchen: Sagenhaft Der Zauberlehrling

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Just as die Verletzung means "the injury" and der Schaden means "the damage," der Schmerz is the main German word for pain. Like weh tun, the verb schmerzen is typically used to describe physical pain.

Da Lisas Fuß am Abend immer noch sehr schmerzte, entschlossen wir uns, die Nacht an diesem schönen Platz [entschlossen, Infinitiv: entschließen] unter Bäumen zu verbringen.

Since Lisa's foot still hurt a lot in the evening, we decided to spend the night in that beautiful place [decided, infinitive: to decide] under some trees.

Captions 11-13, Deutsch mit Eylin: Urlaub im Präteritum

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Further Learning
You'll find many examples like this on Yabla German.

In German, der Klang (the sound) and das Geräusch (the noise) are the words for individual sounds, while words such as der Krach or der Lärm describe general noise that people are bothered by.  

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Let's look at the words for certain types of sounds and noises in German. Because there are many words, today we'll only look at sounds that occur in nature, including animal sounds. Next week, we'll look at sounds produced by humans and machines. There is, of course, some overlap. 

For example, humans and bees can both hum, and humans whistle too, of course. 

Das Geräusch erinnert an das Summen einer Biene.

The sound is reminiscent of the buzz of a bee.

Caption 26, Deutsch mit Eylin: s, ß und ss

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Pfeifen, Zwitschern, Tirilieren

Whistling, chirping, trilling

Caption 4, Kinderlieder zum Mitsingen: Alle Vögel sind schon da

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Here are some examples with the common verbs quietschen, piepen, rascheln, and rauschen.

Während die einen mutig ihr Spielzeug aus dem Wasser retten, hoffen die anderen, dass die Bälle und Quietscheknochen durch ihr Bellen auf magische Art und Weise an den Rand zurückkommen.

While some bravely rescue their toys from the water, the others hope that the balls and squeaky bones will, through their barking, magically come back to the edge.

Captions 3-5, Rheinmain im Blick: Hundeschwimmen

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Mäuse kenne ich. Sie wohnen in Erdlöchern und piepen.

Mice I know. They live in burrows and squeak.

Caption 11, Piggeldy und Frederick: Katze

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Es raschelte manchmal außerhalb des Zeltes,und der Wind rauschte laut in den Bäumen. 

Sometimes there was a rustle outside the tent, and the wind rustled loudly in the trees. 

Captions 20-21, Deutsch mit Eylin: Urlaub im Präteritum

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You may have noticed that the nouns that refer to these sounds are mostly nominalized verbs. These are then neuter. Once you know the noun, it's not at all difficult to remember the related verb and adjective/adverb. 

Rauschend und gurgelnd vereinigen sich viele kleine Rinnsale zu immer größeren Bächen.

Rushing and gurgling, many small brooks combine into increasingly larger streams.

Caption 7, Alpenseen Kühle: Schönheiten

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Die Mutter bat den Bären hinein und der legte sich vor das knisternde Feuer.

The mother asked the bear in and he lay down in front of the crackling fire.

Caption 17, Märchen: Sagenhaft Schneeweißchen und Rosenrot

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Further Learning
You can find more examples of these words on Yabla German, and also look up a few more, such as das Zirpen, das Plätschern, das Brutzeln, or das Knarren.