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Chapter 14: Problem 75

\(\bullet\) 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.

Short Answer

Expert verified
The minimum height is approximately 340.37 meters.

Step by step solution

01

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.
02

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 \( h \).
03

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 \( 334,000 \text{ J/kg} \).
04

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 \( g = 9.81 \text{ m/s}^2 \): \[ h = \frac{0.01 \times 334,000}{9.81} \approx 340.37 \text{ m} \]
05

Conclude the Calculation

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攖hanks 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.

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Most popular questions from this chapter

\(\cdot\) You are given a sample of metal and asked to determine its specific heat. You weigh the sample and find that its weight is 28.4 \(\mathrm{N}\) . You carefully add \(1.25 \times 10^{4} \mathrm{J}\) of heat energy to the sample and find that its temperature rises 18.0 \(\mathrm{C}^{\circ} .\) What is the sample's specific heat?

. Basal metabolic rate. The energy output of an animal engaged in an activity is called the basal metabolic rate (BMR) and is a measure of the conversion of food energy into other forms of energy. A simple calorimeter to measure the BMR consists of an insulated box with a thermometer to measure the temperature of the air. The air has a density of 1.29 \(\mathrm{kg} / \mathrm{m}^{3}\) and a specific heat capacity of 1020 \(\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K}) .\) A 50.0 \(\mathrm{g}\) hamster is placed in a calorimeter that contains 0.0500 \(\mathrm{m}^{3}\) of air at room temperature. (a) When the hamster is running in a wheel, the temperature of the air in the calorimeter rises 1.8 \(\mathrm{C}^{\circ}\) per hour. How much heat does the running hamster generate in an hour? (Assume that all this heat goes into the air in the calorimeter. Neglect the heat that goes into the walls of the box and into the thermometer, and assume that no heat is lost to the surroundings.) (b) Assuming that the hamster converts seed into heat with an efficiency of 10\(\%\) and that hamster seed has a food energy value of 24 \(\mathrm{J} / \mathrm{g}\) , how many grams of seed must the hamster eat per hour to supply the energy found in part (a)?

\(\bullet\) (a) On January \(22,1943,\) the temperature in Spearfish, South Dakota, rose from \(-4.0^{\circ} \mathrm{F}\) to \(45.0^{\circ} \mathrm{F}\) in just 2 minutes. What was the temperature change in Celsius degrees and in kelvins? (b) The temperature in Browning, Montana, was \(44.0^{\circ} \mathrm{Fon}\) January \(23,1916,\) and the next day it plummeted to \(-56.0^{\circ} \mathrm{F} .\) What was the temperature change in Celsius degrees and in kelvins?

\(\bullet\) A laboratory technician drops an 85.0 g solid sample of unknown material at a temperature of \(100.0^{\circ} \mathrm{C}\) into a calorimeter. The calorimeter can is made of 0.150 \(\mathrm{kg}\) of copper and contains 0.200 \(\mathrm{kg}\) of water, and both the can and water are initially at \(19.0^{\circ} \mathrm{C}\) . The final temperature of the system is measured to be \(26.1^{\circ} \mathrm{C}\) . Compute the specific heat capacity of the sample. (Assume no heat loss to the surroundings.)

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?
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