湧く: water from rocks and courage after good news
Compare cold water springing from rocks with courage arising after good news in 湧く. Identify what emerges without replacing the water with boiling or the feeling with a completed brave action.
Cold water springing from between rocks
湧く (わく) can describe water springing forth. 「岩の間から冷たい水が湧いている。」 means “Cold water is springing forth from between the rocks.” 冷たい水 names what emerges, and 岩の間から identifies where it comes from.
湧いている presents the water's continuing flow. Turning it into water becoming hot and boiling would lose both 冷たい and the movement from between the rocks. Start with the water and its source rather than replacing the scene with a similar-sounding word.
Courage suddenly arises after good news
「良い知らせを聞いて、急に勇気が湧いた。」 means “The good news suddenly gave me courage.” 良い知らせを聞いて is the occasion, and 勇気が湧いた describes the change in feeling. What arises here is courage rather than water.
急に adds the suddenness of that change. Courage arising does not itself report a completed brave action. The sentence does not say what challenge the person attempted or whether any such attempt succeeded.
Separate emergence from a later action
The water is springing from between rocks, while the courage arises after good news. Finding 冷たい水 or 勇気 as the subject of 湧く distinguishes a physical flow from the appearance of a feeling. Each example also names a different source or occasion.
The water sentence presents an ongoing state; the courage sentence presents a sudden change. Adding boiling to the first or a successful action to the second would create another story. Read what emerges, where it comes from, or what prompts it using the sentence's own clues.
Does 「冷たい水が湧いている」 mean that the cold water is boiling?
Show the answer and why
No. Cold water is springing from between the rocks. In the other 湧く example, good news gives rise to courage; neither boiling water nor a completed brave action is stated.