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1. To continue driving. How much longer should we drive on before stopping for the night?
2. To motivate one to persist in some task or goal. In this usage, a noun or pronoun can be used between "drive" and "on." The desire to set a good example for my kids drove me on to finally complete my college degree. What drives on this character to survive?
drive someone on (to something)
to make someone move onward toward some kind of success. She said her parents drove her on to finish law school. They drove on their daughter to great things. The thought of earning a large salary drove him on.
to continue driving; to continue with one's journey. We drove on for a little while. The traffic jam is breaking up, so we can drive on.