While painting the top of an antenna 225 m in height, a worker accidentally lets a 1.00-L water bottle fall from his lunchbox. The bottle lands in some bushes at ground level and does not break. If a quantity of heat equal to the magnitude of the change in mechanical energy of the water goes into the water, what is its increase in temperature?

Verified step by step guidance

1. First, identify the initial and final states of the water bottle. Initially, the bottle is at a height of 225 m, and finally, it is at ground level. The change in mechanical energy is due to the change in gravitational potential energy.
2. Calculate the change in gravitational potential energy using the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Δ</mi><mi>E</mi><mo>=</mo><mi>m</mi><mi>g</mi><mi>h</mi></mrow></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is the mass of the water, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is the acceleration due to gravity (approximately 9.81 m/s²), and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> is the height (225 m).
3. Convert the volume of the water bottle (1.00 L) to mass. Since the density of water is approximately 1000 kg/m³, the mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> can be calculated as <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>m</mi><mo>=</mo><mn>1.00</mn><mo>&#xA0;</mo><mi>L</mi><mo>&#xA0;</mo><mo>&#xD7;</mo><mo>&#xA0;</mo><mn>1000</mn><mo>&#xA0;</mo><mi>kg</mi><mo>/</mo><mi>m</mi><mo>³</mo></mrow></math>.
4. Use the formula for heat transfer to find the increase in temperature: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Q</mi><mo>=</mo><mi>m</mi><mi>c</mi><mi>Δ</mi><mi>T</mi></mrow></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math> is the heat transferred (equal to the change in mechanical energy), <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> is the specific heat capacity of water (approximately 4186 J/kg°C), and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Δ</mi><mi>T</mi></math> is the change in temperature.
5. Solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Δ</mi><mi>T</mi></math> using the equation: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Δ</mi><mi>T</mi><mo>=</mo><mfrac><mi>Q</mi><mrow><mi>m</mi><mi>c</mi></mrow></mfrac></mrow></math>. Substitute the values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> to find the increase in temperature.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula U = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height above the reference point. In this scenario, the potential energy of the water bottle at the top of the antenna is converted into other forms of energy as it falls.

Conservation of Energy

The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. As the water bottle falls, its gravitational potential energy is converted into kinetic energy and, upon impact, into thermal energy. Understanding this transformation is crucial to determining the heat absorbed by the water and its subsequent temperature increase.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. It is expressed in joules per kilogram per degree Celsius (J/kg°C). To find the temperature increase of the water, the heat gained from the conversion of mechanical energy must be divided by the product of the water's mass and its specific heat capacity.