[Nagareboshi]

When the verb is in the dictionary form, there's no long vowel at the end. However, when verbs are in the volitional form there is a long vowel at the end. 食べよう、書こう are examples of volitional verbs in the casual/short form. You would use a volitional verb to suggest a plan of action.

昼ご飯を食べよう "Let's eat lunch".

手紙を書こう "Let's write a letter".

Here are the same examples in the polite form:

昼ご飯を食べましょう

手紙を書きましょう