Freezing cold. Nastily I still have to do the laundry by hand. And to keep my hands from being frozen I have to add some hot water to the freezing tap water. Let's just suppose that I have the same amount of both. So the question is, should I add the hot water to the cold or add the cold to the hot, to make the final mixture as warm as possible? Which one will cause less heat loss? Any advice based on either life experience or physical analysis is appreciated. Thanks in advance.
If you are mixing both within a short time then it probably makes no noticeable difference.
If you are first tapping a half a bucket of hot water and then very slowly drip in the cold water then you will have rather hot mixture for a long time which will loose more heat to the environment than if if you do it the other way around.
I am pretty sure you can measure this quite easily if it takes half an hour to fill half a bucket, but for any normal speed (say using a regular water tap): Meh!
Adding the cold water to the hot will result in a warmer mix.
Heat radiates according to the Stefan–Boltzmann law, and the rate of heat loss is proportional to the forth power of the temperature. The quicker you can reduce the temperature of the hot water, the less heat will be lost.
If you add hot water to the cold, halfway through the process, you will have half the hot water left, still radiating quickly. If you add the cold water to the hot, the temperature of the hot water is reduced more quickly.
A variation of this problem is as follows: You drink your coffee after 5 minutes. To ensure you your coffee is as hot as possible, short you add cream immediately, or at the 5 minute mark?
If you add your cream immediately, heat loss from the cooler coffer will be less that if you add the cream after 5 minutes.