James Joule once attempted to measure the increase in temperature of the water in a waterfall resulting from its decrease in gravitational potential energy. How high would the waterfall have to be for the temperature at the bottom to be 1°C higher than at the top?
427 m
To find the height of the waterfall required for the water at the bottom to be 1°C higher than at the top, we can use the principle of conservation of energy.
The increase in temperature of the water can be calculated based on the change in gravitational potential energy:
ΔU = mgh
Where:
ΔU is the change in potential energy
m is the mass of the water
g is the acceleration due to gravity
h is the height of the waterfall
The increase in temperature can be calculated using the formula:
ΔT = ΔU / (m * C)
Where:
ΔT is the change in temperature
C is the specific heat capacity of water
In this case, we want the change in temperature (ΔT) to be 1°C. We know that the specific heat capacity of water (C) is approximately 4.18 J/g°C.
Now, we can rearrange the equation to solve for the height of the waterfall (h):
h = ΔT * (m * C) / (m * g)
Since the mass of the water (m) cancels out, the equation simplifies to:
h = ΔT * C / g
Substituting the known values, we get:
h = 1°C * 4.18 J/g°C / 9.8 m/s²
Simplifying further, we find:
h ≈ 0.427 meters
Therefore, the height of the waterfall would need to be approximately 0.427 meters for the water at the bottom to be 1°C higher than at the top.