# Temperature problem

1. May 30, 2007

### neelakash

1. The problem statement, all variables and given/known data

Two friends A and B receive cups of coffee those are at equal temperature.A adds some cool cream immediately and takes his coffee 10minutes later.B waits for 10 minutes,and adds the same amount of cool cream and begins to take.Decide who takes hotter coffee?

2. Relevant equations
3. The attempt at a solution

2. May 30, 2007

### neelakash

Assuming that the cool cream is cooler than room temperature, friend B takes the hotter coffee.

Newton's law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the temperature of its surroundings.

The rate of change of temperature when the cream is immediately added has 10 minutes worth of effect on the temperature, combined with the temperature of its surroundings.

The rate of change of temperature has no time have much effect on the temperature of friend B's coffee, so only the surroundings' temperature cooled the coffee.

3. May 30, 2007

### Staff: Mentor

Right. The greatest difference leads to the greatest heat loss. So which situation gives the coffee the greatest difference in temperature with its surroundings?

4. May 30, 2007

### lalbatros

Does the solution not depend on the quantity of cream, if it is colder than the room temperature ?
Indeed, we could ask the (apparently) reverse problem:

Two friends receive each an ice cream. A adds a bit of hot cofee on his ice cream, ....

5. May 30, 2007

### Staff: Mentor

I would assume that the cream is sitting in the refrigerator, waiting to be used when needed. (I suppose room temp cream would work just as well.)

6. Jun 4, 2007

### aiy1tm

No, the ice cream question runs into phase transition issues. (ie: if you don't completely melt the ice cream, it will still have a temperature less than or equal to the melting point of ice cream.)

Anyway, it appears the answer to the question is the person who waited 10 minutes, then added cream. If you assume they both add the same amount of cold cream to the same amount of coffee.

The coffee that is not cooled earlier gets to radiate heat at a higher rate for the ten minutes, then it loses a fixed amount of energy to equilibriate the cold cream to whatever temperature.