Problem 63 - College Physics

Question

An \(8.50 \mathrm{~kg}\) 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) 0.000795 kg of ice melts; (b) maximum 0.001432 kg of ice melts.

Step 1: Determine the kinetic energy change

Calculate the change in kinetic energy of the ice block as its speed decreases from 15.0 m/s to 10.0 m/s. The kinetic energy \( KE \) is given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the speed. \[KE_i = \frac{1}{2} \times 8.50 \times (15.0)^2 = 956.25 \text{ J}\]\[KE_f = \frac{1}{2} \times 8.50 \times (10.0)^2 = 425.0 \text{ J}\]Calculate the change in kinetic energy:\[\Delta KE = KE_i - KE_f = 956.25 - 425 = 531.25 \text{ J}\]

Step 2: Calculate the heat absorbed by the ice

Since half of the heat generated goes into the ice, the heat absorbed by the ice \( Q \) is half of the change in kinetic energy:\[Q_{ice} = \frac{1}{2} \times 531.25 = 265.625 \text{ J}\]

Step 3: Determine the mass of ice melted

Use the heat absorbed to find the mass of ice melted. The heat required to melt ice is given by \( Q = mL_f \), where \( m \) is the mass and \( L_f \) is the latent heat of fusion of ice (\( 334,000 \text{ J/kg} \)). Solve for \( m \):\[m = \frac{Q_{ice}}{L_f} = \frac{265.625}{334,000} \approx 0.000795 \text{ kg}\]

Step 4: Calculate maximum kinetic energy converted to heat

To find the maximum amount of ice that could melt, assume all initial kinetic energy of the ice block is converted to heat. The maximum heat available is equal to the initial kinetic energy, which is \( 956.25 \text{ J}\). Again, only half of this goes into the ice:\[Q_{max, ice} = \frac{1}{2} \times 956.25 = 478.125 \text{ J}\]

Step 5: Determine maximum ice melt

Calculate the maximum mass of ice that could melt using the maximum heat absorbed:\[m_{max} = \frac{Q_{max, ice}}{L_f} = \frac{478.125}{334,000} \approx 0.001432 \text{ kg}\]

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. In physics, it is a fundamental concept because it helps us understand how forces and movements influence energy transformation. To calculate the kinetic energy (\( KE \)) of an object, you use the formula:\[ KE = \frac{1}{2}mv^2 \]where:

• \( m \) is the mass of the object in kilograms
• \( v \) is the velocity of the object in meters per second

In our exercise, the ice block changes speed from 15.0 m/s to 10.0 m/s, leading to a change in its kinetic energy. This change indicates that some energy was transferred to another form or another object, revealing the remarkable ways energy can shift within systems. Such transformations are key in understanding broader thermodynamic processes.

Latent Heat of Fusion

Latent heat of fusion is the amount of energy needed to change a substance from solid to liquid without changing its temperature. For ice, this is a well-known value used in calculations. The latent heat of fusion of ice is approximately 334,000 J/kg. This means it takes 334,000 joules to melt one kilogram of ice at 0°C, a crucial aspect when measuring energy absorbed during phase changes. When we calculated how much ice melted due to the heat obtained from slowing down, we used this value to determine the melted mass. Understanding latent heat is vital in calculating how different factors, such as heat transfer and initial kinetic energy, affect the state of matter.

Heat Transfer

Heat transfer is the process of energy moving from one body or place to another and can occur through conduction, convection, or radiation. In our problem, specifically, heat transfer is seen as energy is transferred from the kinetic energy of ice into the internal energy of the ice itself and the floor. Conduction is likely occurring here, as the rough surface of the floor facilitates heat moving from the ice into the floor. The amount of energy absorbed by the ice is summarized in our computations, where only half of the energy loss due to kinetic energy is absorbed by the ice for melting. Understanding this concept helps make sense of how different substances interact thermodynamically and how energy can be conserved or lost in the process.

Physics Problem Solving

Physics problem-solving involves breaking down problems into smaller, manageable parts and applying known principles to arrive at solutions. Our problem-solving steps included identifying the variables involved, calculating kinetic energy changes, and using the latent heat of fusion formula. Some steps you can take while solving similar problems include:

• Clearly outline known values and formulas necessary for calculations
• Transpose and manipulate formulas to solve for desired unknowns
• Check units for consistency to ensure accuracy in final results

By approaching physics problems methodically, as seen in this exercise, you can more easily navigate complex questions and achieve comprehensible solutions. Remember, practicing multiple problems strengthens your skills and understanding of physics concepts more broadly.