Solving Physics Problem: Hot & Cold Liquids w/Formulas

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In summary, the problem asks to find the initial temperatures of hot tea and cold milk when mixed in two different ratios and resulting in two different temperatures. Using the given formula, we can set up two equations and solve for the initial temperatures using algebraic methods.
  • #1
Zygotic Embryo
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Hello, I am just starting Physics. ( Sophmore in high school)

I need some help, understanding and solving this problem.


When you mix hot and cold liquids you can find the temperature of the mixture by using the formula T= ah+bc divided by a+b, where T is the temperature of the mixture, h is the temperature of the hot liquid, c is the temperature of the cold liquid, a and b respresent the amounts of hot and cold liquids. Suppose you mix hot tea and cold milk in a ratio a:b of 9:1 and find that the temperature of the mixture is 117degrees You then change the trea:milk ratio to 2:1 and the temperature drops 96degrees. Find the initial temperatures of the tea and Milk.

Again, I can solve it Logically. But using Formula's ( showing your work ) I am not very found of..

Can someone show how to go about this. Thanks!
 
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  • #2
Can't you solve it using simple algebra?

For your first ratio, you would just plug in your values, and it would look something like this:

117=(9h/10)+(c/10)
for the second:
96=(2h/3)+(c/3)

Simplify them and work them out separately, until you have both of the variables (h and c) on one side for both equations.

You can then have "h" equal something ±c, then in the second equation inject what h equals into h's place. Solve the equation for c, then solve the first equation as you would a one-variable algebraic equation.
 
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  • #3


Hello, welcome to the world of physics! It's great that you are starting to learn about this fascinating subject. Let's dive into this problem and see how we can solve it using formulas.

First, let's start with the given information: the ratio of hot tea to cold milk is 9:1, and the temperature of the mixture is 117 degrees. We can represent this using the formula T= ah+bc/a+b. Plugging in the values, we get:

117 = (9h + c)/10

Next, we are asked to change the ratio to 2:1 and the temperature drops to 96 degrees. We can set up a second equation using the same formula:

96 = (2h + c)/3

Now, we have two equations and two unknowns (h and c). We can solve this system of equations using algebra. First, let's multiply the first equation by 3 to get rid of the fractions:

351 = 27h + 3c

Next, let's multiply the second equation by 10 to get rid of the fractions:

960 = 20h + 10c

Now, we have two equations with the same variables, so we can subtract them to eliminate one variable:

351 - 960 = 27h - 20h + 3c - 10c

-609 = 7h - 7c

Finally, we can solve for one variable in terms of the other:

h = (7c + 609)/7

Now, we can substitute this into one of the original equations to solve for c. Let's use the second equation:

96 = (2(7c+609)/7 + c)/3

Simplifying this, we get:

96 = (14c + 609 + 3c)/21

96 = (17c + 609)/21

Now, we can solve for c:

1616 = 17c + 609

1007 = 17c

c = 59.24 degrees

Now, we can plug this value back into our equation for h to solve for h:

h = (7(59.24) + 609)/7

h = 95.06 degrees

So, the initial temperatures of the tea and milk were approximately 95.06 degrees and 59.24 degrees, respectively.

I understand that using formulas may seem daunting, but they are a
 

FAQ: Solving Physics Problem: Hot & Cold Liquids w/Formulas

What is the formula for calculating the change in temperature of a hot or cold liquid?

The formula for calculating the change in temperature is ΔT = Tf - Ti, where ΔT is the change in temperature, Tf is the final temperature, and Ti is the initial temperature.

How do you calculate the specific heat capacity of a liquid?

The specific heat capacity of a liquid is calculated using the formula c = Q/mΔT, where c is the specific heat capacity, Q is the heat energy absorbed or released, m is the mass of the liquid, and ΔT is the change in temperature.

Can you provide an example of solving a hot and cold liquid problem using the formula?

For example, if 500g of water at 20°C is mixed with 300g of water at 80°C, what is the final temperature of the mixture? Using the formula ΔT = Tf - Ti, we can calculate the change in temperature of the hot water as 80-20 = 60. Then, using the formula c = Q/mΔT and assuming the specific heat capacity of water is 4.186 J/g°C, we can calculate the heat energy released by the hot water as Q = (300g)(4.186 J/g°C)(60°C) = 75,228 J. Finally, using the formula Tf = Q/mc + Ti, we can calculate the final temperature of the mixture as Tf = (75,228 J)/(500g)(4.186 J/g°C) + 20°C = 55.05°C.

How do you account for the latent heat of fusion or vaporization in hot and cold liquid problems?

In cases where a liquid undergoes a phase change, the formula c = Q/mΔT is modified to account for the latent heat of fusion or vaporization. For example, if a 100g ice cube at -20°C is added to 200g of water at 20°C, the final temperature of the mixture can be calculated using the formula Tf = (m1c1T1 + mL + m2c2T2)/(m1c1 + mL + m2c2), where m1 and c1 represent the mass and specific heat capacity of ice, mL represents the latent heat of fusion, m2 and c2 represent the mass and specific heat capacity of water, and T1 and T2 represent the initial temperatures of the ice and water, respectively.

What are some common mistakes to avoid when solving hot and cold liquid problems?

Some common mistakes to avoid include using the wrong units for mass or heat energy, not accounting for the specific heat capacity of each liquid, and not accounting for the latent heat of fusion or vaporization if applicable. It is also important to ensure that the initial and final temperatures are in the correct order when using the formula ΔT = Tf - Ti.

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