Learn Hardcore German: Die Laterne wirft einen langen Schatten auf den Sand des Spielplatzes. - The lantern casts a long shadow on the sand of the playground.

Die Laterne is the subject of the verb wirft. In German, the subject of a sentence always stands in the nominative case. Since Laterne is a feminine noun, its singular nominative article is die.

Why does einen langen Schatten use the accusative case, and how do the article and adjective endings work here?

Einen langen Schatten is the direct object of wirft (what the lamp throws). Direct objects take the accusative.

einen is the masculine singular accusative indefinite article.
lang gets the weak adjective ending -en after an article that already shows case (here einen).
Schatten is a masculine noun whose accusative singular form is identical to its nominative: Schatten.

What does einen Schatten werfen really mean, and isn’t werfen just “to throw”?

Literally werfen means “to throw,” but in the idiom einen Schatten werfen it translates as “to cast a shadow.” It’s a common German collocation where the action of “throwing” becomes “casting.”

Why does the verb werfen change to wirft in this sentence?

Werfen is a stem-changing verb. In the present tense 3rd person singular (er/sie/es), the e in the stem changes to i, and you add -t. So:

ich werfe
du wirfst
er/sie/es wirft

Why is auf followed by den Sand (accusative), and when would you use dative instead?

Auf is a two-way (Wechsel-) preposition.

Use accusative when it indicates movement toward or onto something (“throw a shadow onto the sand”). Hence auf den Sand.
Use dative for static location (“the lamp stands on the sand” → auf dem Sand).

Why is des Spielplatzes in the genitive case, and could you use vom Spielplatz instead?

Des Spielplatzes shows possession: “the sand of the playground.” That’s genitive singular for a masculine noun. You can replace a genitive with von + dative in everyday speech:

des Spielplatzes → vom Spielplatz
Both mean “of/from the playground,” though des Spielplatzes is slightly more formal or literary.

Why does Spielplatz take -es in the genitive singular (Spielplatzes)?

Masculine (and neuter) nouns normally form the genitive singular with -s or -es. Short, one-syllable nouns often add -es for easier pronunciation. Hence Spielplatz → Spielplatzes.

Could you change the word order, and if so, what would that emphasize?

Yes. German follows the verb-second (V2) rule: any sentence element can come first for emphasis. For example:
Einen langen Schatten wirft die Laterne auf den Sand des Spielplatzes.
This front-loads einen langen Schatten, stressing how long or how prominent the shadow is, while keeping wirft in second position.