Problem 75 - College Physics

Question

\(\bullet\) A piece of ice at \(0^{\circ} \mathrm{C}\) falls from rest into a lake whose tem- perature is \(0^{\circ} \mathrm{C},\) and 1.00\(\%\) of the ice melts. Compute the minimum height from which the ice has fallen.

Step-by-Step Solution

Answer

The minimum height is approximately 340.37 meters.

Step 1: Identify the Energy Conversion

When the ice falls, its potential energy is converted into kinetic energy, which then causes some of the ice to melt. This melting absorbs energy in the form of the heat of fusion.

Step 2: Establish the Potential Energy Equation

The potential energy converted to thermal energy is given by \( mgh \), where \( m \) is the mass of the ice, \( g \) is the gravitational acceleration (\( 9.81 \text{ m/s}^2 \)), and \( h \) is the height. We need to compute \( h \).

Step 3: Determine the Energy for Melting Ice

Since 1% of the ice melts, the energy absorbed by the melting ice is given by \( 0.01mL_f \), where \( L_f \) is the latent heat of fusion for ice, approximately \( 334,000 \text{ J/kg} \).

Step 4: Equate the Energies and Solve for Height

The potential energy \( mgh \) is equal to the energy required to melt 1% of the ice, \( 0.01mL_f \). Form the equation: \( mgh = 0.01mL_f \). Simplify and solve for \( h \): \[ gh = 0.01L_f \] \[ h = \frac{0.01L_f}{g} \] Substituting \( L_f = 334,000 \text{ J/kg} \) and \( g = 9.81 \text{ m/s}^2 \): \[ h = \frac{0.01 \times 334,000}{9.81} \approx 340.37 \text{ m} \]

Step 5: Conclude the Calculation

The minimum height from which the ice must have fallen to melt 1% of it is approximately 340.37 meters.

Key Concepts

Potential Energy

When an object is in a position where it can fall, it has potential energy. This energy is due to its position relative to the Earth. The higher the object is, the more potential energy it has. In the case of the ice falling into the lake, this potential energy can be calculated using the formula:
\( PE = mgh \),
where:

• \( m \) is the mass of the ice,
• \( g \) is the acceleration due to gravity \( (9.81 \text{ m/s}^2) \),
• \( h \) is the height from which the ice falls.

As the ice falls, this energy is converted into other forms. Understanding this concept helps us calculate the height needed for 1% of the ice to melt upon impact.

Kinetic Energy

Once the ice begins to fall, its potential energy starts converting into kinetic energy. Kinetic energy is the energy of motion.
It is expressed by the formula:
\( KE = \frac{1}{2}mv^2 \).
For the ice chunk, as it hits the surface of the lake, all the potential energy it once had is now mostly in the form of kinetic energy.

• The transformation stops when the ice hits the lake.
• The kinetic energy at this moment is what contributes to melting some of the ice.

This whole process is why only 1% of the ice melts—thanks to energy conservation.

Heat of Fusion

The heat of fusion, or the enthalpy of fusion, is the energy needed to change a substance from a solid to a liquid at its melting point, without changing its temperature.

• For ice, this is approximately \( 334,000 \text{ J/kg} \).
• It is the energy required to overcome the forces holding the water molecules in a solid structure, allowing the ice to melt.

In the problem, 1% of the ice melts, which means that a small portion of the energy from the fall is used in increasing the ice's internal energy to initiate the phase change.
Knowing the heat of fusion helps us determine how much energy is needed to melt just a small fraction of the ice.

Latent Heat

Latent heat is the energy absorbed or released during a phase change of a substance, occurring without a change in temperature.

• In the context of the problem, the latent heat involved is the latent heat of fusion.
• This energy is what leads ice to melt into water at \( 0^{\circ} \text{C} \).

Understanding latent heat allows us to comprehend why, despite the ice and the water being at the same temperature, some of the ice melts upon impact.
It absorbs the energy that once was in the form of kinetic energy, making a small percentage of the ice transition into a liquid state.