When you do something in a forceful or intense way, you do it vigorously. If you're trying to figure out which of your brothers ate the last of the cookies, it's probably the one who's denying it most vigorously.
The adverb vigorously has the English root "vigor," which comes from the Latin word vigere, meaning to be lively or flourish. Synonyms for vigorously include energetically, forcefully, and strenuously. If you're trying to lose weight, you should eat fewer cookies and walk vigorously every day, and if you want to get your dishes really clean, you'll have to scrub them vigorously.