A certain mass of water at the top of Niagara Falls has a temperature of +10.0 degrees Celsius. It falls a distance of 50m. Assuming that all its potential energy is converted into thermal energy, calculate the temperature of the mass of water at the bottom of the falls.
m g h = m C delta T
so
delta T = g h /C
where C is specific heat of water.
To determine the temperature of the mass of water at the bottom of the falls, we can use the principle of conservation of energy. The potential energy of the water at the top of the falls is converted entirely into thermal energy at the bottom.
First, we need to calculate the change in potential energy of the water as it falls a distance of 50m. The formula for potential energy is:
Potential energy = mass × gravitational acceleration × height
In this case, the height is 50m and the gravitational acceleration is approximately 9.8 m/s².
Next, we need to convert the change in potential energy into thermal energy. The thermal energy is given by the formula:
Thermal energy = mass × specific heat capacity × change in temperature
The specific heat capacity of water is approximately 4.18 J/g°C.
Since the question provides the initial temperature of the water at the top of the falls and we want to find the final temperature at the bottom, we can rearrange the formula above to solve for the change in temperature:
Change in temperature = Thermal energy / (mass × specific heat capacity)
Given that the mass of water is not provided, we cannot calculate the exact final temperature. We need to know the mass of the water in order to calculate the change in temperature.
To calculate the temperature of the mass of water at the bottom of the falls, we can use the principle of conservation of energy.
The potential energy (PE) of the water at the top of the falls is given by:
PE = m * g * h
where m is the mass of the water, g is the acceleration due to gravity (assumed to be 9.8 m/s^2), and h is the height of the falls (50m).
The thermal energy (TE) gained by the water at the bottom is given by:
TE = m * c * ΔT
where c is the specific heat capacity of water (approximately 4.18 J/g°C), and ΔT is the change in temperature.
Since all the potential energy is converted into thermal energy, we can equate the two expressions:
PE = TE
m * g * h = m * c * ΔT
Simplifying the equation:
g * h = c * ΔT
ΔT = (g * h) / c
Substituting the given values:
ΔT = (9.8 m/s^2 * 50m) / 4.18 J/g°C
ΔT ≈ 117.7 °C
Therefore, the temperature of the mass of water at the bottom of the falls would be approximately +10.0 °C + 117.7 °C = 127.7 °C.