Problem 73 - College Physics

Question

A \(0.4 \mathrm{~kg}\) piece of ice at \(-10^{\circ} \mathrm{C}\) is dropped from a height \(h .\) Upon impact, \(5 \%\) of its kinetic energy is converted into heat energy. If the impact transforms all of the ice into water that has a final temperature of \(0^{\circ} \mathrm{C},\) find \(h\)

Step-by-Step Solution

Answer

The ice must be dropped from a height of approximately 7234.6 meters.

Step 1: Determine the Energies Involved

To solve this problem, we'll use both potential and kinetic energy concepts. First, understand that the potential energy lost by the ice when it falls is converted into kinetic energy. Upon impact, some kinetic energy is converted into heat, melting the ice. Let's denote the height from which the ice is dropped as \( h \), and its mass as \( m = 0.4 \text{ kg} \). The specific heat of ice is \( c_{ice} = 2.1 \text{ J/g°C} \), the heat of fusion for ice \( L_f = 334 \text{ J/g} \), and the acceleration due to gravity is \( g = 9.81 \text{ m/s}^2 \).

Step 2: Calculate the Energy Required to Raise the Temperature of Ice

Since the ice is initially at \(-10^{\circ}\text{C}\), we need to calculate the energy required to bring the temperature of ice to \(0^{\circ}\text{C}\) before it can start melting. This energy (\( Q_1 \)) is given by \( Q_1 = mc_{ice}\Delta T \), where \( \Delta T = 10 ^{\circ}\text{C} \). Substituting the values: \( Q_1 = 0.4 \times 1000 \times 2.1 \times 10 = 8400 \text{ J} \). Note that \( 0.4 \text{ kg} = 400 \text{ g} \).

Step 3: Calculate the Energy Required to Melt the Ice

The energy needed to melt the ice (\( Q_2 \)) is calculated using the latent heat formula: \( Q_2 = mL_f \), where \( m = 400 \text{ g} \) and \( L_f = 334 \text{ J/g} \). Thus, \( Q_2 = 400 \times 334 = 133600 \text{ J} \).

Step 4: Calculate the Total Energy Required

Add the energies required to raise the temperature and to melt the ice. Total energy required (\( Q_{total} \)) is \( Q_{total} = Q_1 + Q_2 = 8400 + 133600 = 142000 \text{ J} \).

Step 5: Relate the Potential Energy to Kinetic Energy and Solve for Height

Only 5% of the kinetic energy is converted to heat, thus 0.05KE = Q_total. The potential energy lost when the ice falls from height \( h \) is given by \( mgh \). Since potential energy converts to kinetic energy, \( KE = mgh \). Thus, 0.05 \( mgh = 142000 \). Solving for \( h \), we plug in \( m = 0.4 \), \( g = 9.81 \): \(\begin{align*}0.05 \times 0.4 \times 9.81 \times h &= 142000 \h &= \frac{142000}{0.05 \times 0.4 \times 9.81} \h &\approx 7234.6 \text{ m}\end{align*}\)

Step 6: Final Step: Conclusion

Thus, to convert all the ice into water at \(0^{\circ} \mathrm{C}\), the ice must be dropped from a height of approximately \(7234.6 \text{ meters}\).

Key Concepts

Kinetic Energy

Kinetic energy is a form of energy that an object possesses due to its motion. When you drop an object, like the ice in this exercise, its potential energy is transferred into kinetic energy during the fall. The formula to calculate kinetic energy (KE) of an object is given by \[ KE = \frac{1}{2}mv^2 \] where

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

In the scenario, the ice converts its potential energy entirely into kinetic energy as it falls. Upon impact with the ground, part of this kinetic energy is transformed into heat, contributing to the melting of the ice. This concept is important because it demonstrates the energy conversion from motion into other forms, such as heat, which is key for understanding how energy transfers can perform work or change the state of matter.

Potential Energy

Potential energy is the energy stored in an object due to its position. For objects on Earth, this is often gravitational potential energy, which depends on the object's height above the ground. The formula for gravitational potential energy (PE) is \[ PE = mgh \] where

• \( m \) is the mass in kilograms,
• \( g \) is the acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \),
• \( h \) is the height from which the object is released.

In our problem, as the ice is dropped from height, it possesses gravitational potential energy. This energy transforms entirely into kinetic energy as it falls, illustrating a critical principle of energy conservation. In essence, what the object loses in potential energy as it descends is gained in kinetic energy, just before impact.

Heat Transfer

Heat transfer is the movement of thermal energy from one object or substance to another. In our ice melting scenario, when the piece of ice hits the ground, some of its kinetic energy is transformed into thermal energy. This heat energy is then transferred to the ice, raising its temperature. It's crucial to grasp that only a portion of kinetic energy becomes heat, specifically 5%, in this case. Once the ice absorbs the thermal energy, it can increase in temperature and transition from a solid to a liquid. The energy transformation that occurs during impact provides the necessary heat to first raise the ice's temperature to zero degrees Celsius, and then begin the melting process, which is central to understanding how energy operates in physical systems.

Specific Heat Capacity

Specific heat capacity is a property that indicates how much energy is required to change the temperature of a substance. It's defined as the amount of heat per unit mass required to raise the temperature by one degree Celsius. The specific heat capacity is denoted by \( c \) and the formula is \[ Q = mc\Delta T \] where

• \( Q \) is the heat energy added, in joules,
• \( m \) is the mass in kilograms,
• \( c \) is the specific heat capacity of the substance,
• \( \Delta T \) is the change in temperature in degrees Celsius.

For ice, the specific heat capacity helps determine how much energy is needed to bring its temperature from \(-10^{\circ} \text{C} \) to \(0^{\circ} \text{C} \). In our exercise, before melting can occur, there must be enough heat to raise the temperature to the point where phase change can begin. Thus, specific heat capacity is a critical element in understanding energy requirements for temperature changes.