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As long as the water is actually boiling the whole time, then there is no difference whatsoever.

Boiling water can only have one temperature (around $100^\circ\mathrm{C}$100 C, but adjusted somewhat for altitude or impurities in the water), and it stays at that temperature as long as it's in a state of boiling. Keeping the heat up only serves to hasten the transition from water to steam.

The heat transferred into the eggs (or whatever you are cooking) depends only on the temperature of the water, not on the vigor with which it is boiling. As a result, keeping the gas on high will generally have two effects: (1) wasting more gas by heating up the room more and turning more water into steam than necessary, and (2) increasing the chances that the eggs will be tossed around enough to crack. On the other hand, if you turn the gas so low that the water is no longer truly boiling, its temperature will drop below the boiling point, slowing the cooking process (and possibly even changing the end result - cooking is a complicated chemical process that relies on the right amount of heat being applied over the proper length of time).

Of course, there are cases where a recipe may call for a particular level of boiling, but my guess is this usually has to do with keeping the food from sticking to the pot or some such thing, with vigorous boiling replacing the need for constant stirring.

As long as the water is actually boiling the whole time, then there is no difference whatsoever.

Boiling water can only have one temperature (around $100^\circ\mathrm{C}$, but adjusted somewhat for altitude or impurities in the water), and it stays at that temperature as long as it's in a state of boiling. Keeping the heat up only serves to hasten the transition from water to steam.

The heat transferred into the eggs (or whatever you are cooking) depends only on the temperature of the water, not on the vigor with which it is boiling. As a result, keeping the gas on high will generally have two effects: (1) wasting more gas by heating up the room more and turning more water into steam than necessary, and (2) increasing the chances that the eggs will be tossed around enough to crack. On the other hand, if you turn the gas so low that the water is no longer truly boiling, its temperature will drop below the boiling point, slowing the cooking process (and possibly even changing the end result - cooking is a complicated chemical process that relies on the right amount of heat being applied over the proper length of time).

Of course, there are cases where a recipe may call for a particular level of boiling, but this usually has to do with keeping the food from sticking to the pot or some such thing, with vigorous boiling replacing the need for constant stirring.

As long as the water is actually boiling the whole time, then there is no difference whatsoever.

Boiling water can only have one temperature (around 100 C, but adjusted somewhat for altitude or impurities in the water), and it stays at that temperature as long as it's in a state of boiling. Keeping the heat up only serves to hasten the transition from water to steam.

The heat transferred into the eggs (or whatever you are cooking) depends only on the temperature of the water, not on the vigor with which it is boiling. As a result, keeping the gas on high will generally have two effects: (1) wasting more gas by heating up the room more and turning more water into steam than necessary, and (2) increasing the chances that the eggs will be tossed around enough to crack. On the other hand, if you turn the gas so low that the water is no longer truly boiling, its temperature will drop below the boiling point, slowing the cooking process (and possibly even changing the end result - cooking is a complicated chemical process that relies on the right amount of heat being applied over the proper length of time).

Of course, there are cases where a recipe may call for a particular level of boiling, but my guess is this usually has to do with keeping the food from sticking to the pot or some such thing, with vigorous boiling replacing the need for constant stirring.

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As long as the water is actually boiling the whole time, then there is no difference whatsoever.

Boiling water can only have one temperature (around $100^\circ\mathrm{C}$, but adjusted somewhat for altitude or impurities in the water), and it stays at that temperature as long as it's in a state of boiling. Keeping the heat up only serves to hasten the transition from water to steam.

The heat transferred into the eggs (or whatever you are cooking) depends only on the temperature of the water, not on the vigor with which it is boiling. As a result, keeping the gas on high will generally have two effects: (1) wasting more gas by heating up the room more and turning more water into steam than necessary, and (2) increasing the chances that the eggs will be tossed around enough to crack. On the other hand, if you turn the gas so low that the water is no longer truly boiling, its temperature will drop below the boiling point, slowing the cooking process (and possibly even changing the end result - cooking is a complicated chemical process that relies on the right amount of heat being applied over the proper length of time).

Of course, there are cases where a recipe may call for a particular level of boiling, but this usually has to do with keeping the food from sticking to the pot or some such thing, with vigorous boiling replacing the need for constant stirring.