Problem 65 - College Physics

Question

An 8.50 kg block of ice at \(0^{\circ} \mathrm{C}\) is sliding on a rough horizontal icehouse floor (also at \(0^{\circ} \mathrm{C} )\) at 15.0 \(\mathrm{m} / \mathrm{s} .\) Assume that half of any heat generated goes into the floor and the rest goes into the ice. (a) How much ice melts after the speed of the ice has been reduced to 10.0 \(\mathrm{m} / \mathrm{s} ?\) (b) What is the maximum amount of ice that will melt?

Step-by-Step Solution

Answer

(a) 0.00239 kg of ice melts after the speed decreases to 10.0 m/s. (b) The maximum amount of ice that will melt is 0.00143 kg.

Step 1: Understand Energy Changes

First, determine the change in kinetic energy of the ice block as it slows down. The change in kinetic energy, \\( \Delta K \), is given by: \[ \Delta K = \frac{1}{2} m (v_i^2 - v_f^2) \] where \( m \) is the mass, \( v_i \) is the initial velocity, and \( v_f \) is the final velocity.

Step 2: Calculate the Change in Kinetic Energy

Substitute the given values into the kinetic energy change formula:\[ \Delta K = \frac{1}{2} \times 8.50 \, \text{kg} \times (15.0^2 \, \text{m/s} - 10.0^2 \, \text{m/s}) \] \[ \Delta K = \frac{1}{2} \times 8.50 \, \text{kg} \times (225 - 100) \, \text{m}^2/\text{s}^2 \] \[ \Delta K = 12.75 \times 125 \, \text{J} = 1593.75 \, \text{J} \]

Step 3: Determine Energy to Melt the Ice

Only half of the lost kinetic energy goes into melting the ice because the other half goes into the floor. Therefore, the energy used to melt the ice is:\[ E_{melt} = \frac{\Delta K}{2} = \frac{1593.75}{2} = 796.875 \, \text{J} \]

Step 4: Calculate the Mass of Ice Melted

Use the latent heat of fusion for ice, which is 334,000 J/kg, to find the mass of ice melted, \( m_f \):\[ m_f = \frac{E_{melt}}{L_f} = \frac{796.875}{334,000} \, \text{kg} \approx 0.002385 \, \text{kg} \]

Step 5: Determine the Maximum Potential Energy Loss

If the block loses all its kinetic energy, the total initial kinetic energy used to melt the ice is:\[ K_{total} = \frac{1}{2} m v_i^2 = \frac{1}{2} \times 8.50 \, \text{kg} \times 15.0^2 \, \text{m/s} \] \[ K_{total} = 956.25 \, \text{J} \] Half of this energy goes into the melting:\[ E_{max\, melt} = \frac{956.25}{2} = 478.125 \, \text{J} \]

Step 6: Calculate the Maximum Amount of Ice Melted

Finally, use the latent heat of fusion to calculate the maximum mass of ice that can melt:\[ m_{max\, f} = \frac{E_{max\, melt}}{L_f} = \frac{478.125}{334,000} \, \text{kg} \approx 0.001432 \, \text{kg} \]

Key Concepts

Kinetic Energy

When we think of kinetic energy, we are considering the energy that an object possesses due to its motion. It's dependent on two main factors: the mass of the object and its velocity. The formula to calculate kinetic energy, \( K \), is given by

• \( K = \frac{1}{2} m v^2 \)

where \( m \) represents the mass of the object, and \( v \) is its velocity.
In the exercise, the ice block's motion on the floor involves kinetic energy that changes due to friction and results in the ice block slowing down. By calculating the initial and final kinetic energies of the ice block, we determine how much mechanical energy was lost. This lost energy doesn't just disappear; it transforms into other types of energy, like heat.
Thus, understanding kinetic energy not only helps in comprehending the dynamics of motion but is also crucial in examining how energy can be transferred or transformed in physical processes.

Latent Heat of Fusion

Latent heat of fusion is a vital concept in thermodynamics, particularly when involving phase changes in substances. It refers to the amount of energy required to change a substance's state from solid to liquid at a constant temperature and pressure.

• For ice, the latent heat of fusion is 334,000 J/kg.

This energy is required to break the molecular bonds in ice, allowing it to transition into water without changing temperature.
In the context of the exercise, when the ice block slows down, some of its kinetic energy is converted into heat. Part of this heat is absorbed by the ice, providing the energy necessary to melt it. By dividing the energy available for melting by the latent heat of fusion, we determine the mass of the ice that can be melted. Understanding this concept aids in quantifying energy changes involved in melting processes and solidifies comprehension of phase transformations.

Energy Conservation

Energy conservation is a fundamental principle in physics stating that energy cannot be created or destroyed, only transformed or transferred from one form to another. This timeless law implies that the total energy in an isolated system remains constant.

• The initial kinetic energy of the ice eventually converts into thermal energy.
• This thermal energy is then partly transferred to the ice and floor.

In the exercise, when the ice block loses kinetic energy as it slows down, the energy doesn't vanish. Instead, due to the law of conservation, it turns into heat energy. Half of this heat contributes to melting the ice, while the other half is absorbed by the surrounding floor.
Recognizing how energy conservation works helps us track the flow and conversion of energy within the system. It is a foundational concept for analyzing energy transformations, ensuring that calculations in thermodynamics are accurate and reliable.