How Much Warmer Could Water Get After a 50 Meter Fall?

[caesius]

Homework Statement

On his honeymoon, James Joule tested the conversion of mechanical energy into internal energy by measuring temperatures of falling water. If water at the top of a Swiss waterfall has a temperature of 10.0 degrees Celsius and then falls 50.0 m, what maximum temperature at the bottom could Joule expect?

Homework Equations

change in internal energy = mgh

The Attempt at a Solution

I'm sure this question is easy I must be missing something, the change in internal energy is mgh but we have no mass?

The answer is (10.0 + 0.117) but I don't see how this was found.

 

[olgranpappy]

use
[tex]
\Delta E = c M \Delta T
[/tex]
where c is the specific heat. the mass drops out.

 

[Moetasim]

I would like to clarify that the change in internal energy is not simply mgh, but rather, it is the product of mass, gravitational acceleration, and the change in height. In this case, the mass of the falling water and the gravitational acceleration are constant, so the only variable is the change in height.

To find the maximum temperature at the bottom of the waterfall, we can use the equation for change in internal energy (ΔU = mgh) and the fact that the internal energy of a substance is directly proportional to its temperature (U ∝ T). Therefore, we can set up the following proportion:

ΔU1/T1 = ΔU2/T2

Where ΔU1 is the change in internal energy at the top of the waterfall, T1 is the initial temperature (10.0 degrees Celsius), ΔU2 is the change in internal energy at the bottom of the waterfall, and T2 is the maximum temperature we are trying to find.

Since we know that the change in internal energy is mgh, we can rewrite the equation as:

mgh/T1 = mgh/T2

We can then cancel out the mass and gravitational acceleration, leaving us with:

h/T1 = h/T2

Solving for T2, we get:

T2 = (h/T1) * T1

Plugging in the values given in the problem, we get:

T2 = (50.0 m/10.0 degrees Celsius) * 10.0 degrees Celsius

T2 = 50.0 degrees Celsius

Therefore, the maximum temperature at the bottom of the waterfall that Joule could expect is 50.0 degrees Celsius. This calculation assumes that all of the mechanical energy of the falling water is converted into internal energy, which may not be the case due to factors such as friction and energy losses.