Answer:
Judiciously: done or carried out in a wise or sensible way having a sound judgment.
Rewriting the sentence with an appropriate synonym:
His father nodded wisely, "He would know. Why physics?"
Step-by-step explanation:
The sentence means that the father nodded in agreement in a wise or sensible way. This will mean that the father had weighed the options and had good judgment on the matter.
Judiciously is an adverb which answers the question of how the father nodded. Synonyms of the word judiciously are: wisely, sensibly, astutely, etc.