Rearranging equations-need clarifying & pointing in the right direction (long)

[rose22]

Rearranging equations-need clarifying & pointing in the right direction! (long)

1. For an assignment question, I need to rearrange equations to find the temperature increase of a waterfall, specifically energy that has been turned into internal heat of the water. The height of the drop is 365 m. Mass of water is 1,000 kg (though the question states answer can be found without it). There isn't any net transfer of energy between mass of water & kinetic energy gained in the fall.



2. Equations that I've used so far:
ΔE_k = 1/2 mv²
W = mgh

g = constant

I know that I need to use q = mcΔT



3. So far, this is what I have done:

(rearranged from 1/2 mv² = mgh)

v = √2gh
v = 2 x 9.8 x 365 = 7154 (7 x 10³)
√7154 = 84.58 = 85

Then, I did:
(rearranged from q = mcΔT)

ΔT = q divided by mc
85 / 1000 x (4.2 x 10³) = 357

Is this about right? How can I put the 3 equations together, or are they alright like that?

Thank you!

 

[dikmikkel]

q is in fact an energy. It is the energy gained. Do it symbolically first and then insert your values.
How can you insert the velocity in the heating equation ?
You can find the decrease in potential energy which is equal to the gained kinetic energy and then the temperature. Maybe like this:
dE = 0 <=> T = V <=> T = mgh
And plug that into the T = mgh = q = mc dt <=> gh = c dt <=> dt = gh/c
Hope that i understood the problem.

 

[rose22]

i think i understand, but where does the 0 come in?? so, with T = mgh, is that deltaT?

so putting the equations together would result in:

T = mgh = dt = gh/c ?

 

[dikmikkel]

The T stands for kinetic energy, the V potential dE in- or decrease in energy.
The zero is because the energy is conserved(Or all potential energy turns into kinetic/Heat). I think something is missing in the question if it isn't right. The way i understand it is: All potential energy lost turns into heat and hence:
$V = mgh = mc\Delta T\\ \Delta T = \dfrac{gh}{c}$
Where the capital T is temperature.

 

[asteriatic]

Hello,

I can provide some clarification and guidance on rearranging equations to find the temperature increase of a waterfall.

Firstly, it is important to understand the concepts and variables involved in the question. In this case, we are dealing with the energy of a waterfall and its conversion into internal heat of the water. The key variables are the height of the waterfall (h), mass of water (m), and the gravitational constant (g).

Based on the equations you have provided, it seems like you are on the right track. The equations for kinetic energy (ΔEk) and work (W) are applicable in this scenario. However, we also need to consider the specific heat capacity (c) and the change in temperature (ΔT) of the water.

To rearrange the equations, we can start with the equation for work (W = mgh) and substitute the value for gravitational constant (g) to get:

W = mg x h

Next, we can substitute the equation for kinetic energy (ΔEk = 1/2 mv2) into the equation for work, which gives us:

ΔEk = mg x h

Now, we can rearrange the equation to solve for the change in temperature (ΔT):

ΔT = ΔEk / (mc)

Substituting the values given in the question, we get:

ΔT = (1000 kg x 9.8 m/s2 x 365 m) / (1000 kg x 4.2 kJ/kg∙K)

= 85.24 K

Therefore, the temperature increase of the waterfall is approximately 85.24 Kelvin.

In summary, to rearrange equations, it is important to understand the concepts and variables involved and use the appropriate equations to solve for the desired variable. I hope this helps to clarify and guide you in the right direction. Let me know if you have any further questions. Good luck with your assignment!