Problem 55 - University Physics with Modern Physics

Question

An asteroid with a diameter of 10 \(\mathrm{km}\) and a mass of \(2.60 \times 10^{15} \mathrm{kg}\) impacts the earth at a speed of \(32.0 \mathrm{km} / \mathrm{s},\) landing in the Pacific Ocean. If 1.00\(\%\) of the asteroid's kinetic energy goes to boiling the ocean water (assume an initial water temperature of \(10.0^{\circ} \mathrm{C}\) ), what mass of water will be boiled away by the collision? (For comparison, the mass of water contained in Lake Superior is about \(2 \times 10^{15} \mathrm{kg} .\) )

Step-by-Step Solution

Answer

The asteroid boils approximately \(5.04 \times 10^{16} \mathrm{kg}\) of water, surpassing Lake Superior's mass.

Step 1: Calculate the Asteroid's Kinetic Energy

The formula to find the kinetic energy \( KE \) of an object is given by \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity. For the asteroid:\[KE = \frac{1}{2} \times 2.60 \times 10^{15} \, \mathrm{kg} \times (32,000 \, \mathrm{m/s})^2\]Calculating, \[KE = \frac{1}{2} \times 2.60 \times 10^{15} \times 1.024 \times 10^9 \]\[KE = 1.33 \times 10^{25} \, \mathrm{J}\]

Step 2: Determine the Energy Used to Boil Water

Since 1\(\%\) of the asteroid's kinetic energy is used to boil water, calculate that fraction of energy:\[E_{boil} = \frac{1}{100} \times 1.33 \times 10^{25} \, \mathrm{J} = 1.33 \times 10^{23} \, \mathrm{J}\]

Step 3: Calculate Heat Required to Raise Water to Boiling

First, we must calculate the heat required to raise the temperature of water from \(10^{\circ}\mathrm{C}\) to \(100^{\circ}\mathrm{C}\). Use the specific heat formula and the specific heat capacity of water \(c = 4186 \, \mathrm{J/kg \, ^\circ C}\):\[Q_1 = mc\Delta T\]Where \( \Delta T = 100^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C} = 90^{\circ} \mathrm{C}\). The equation becomes:\[Q_1 = 4186 \times 90 \times m\]

Step 4: Calculate Heat for Phase Change from Water to Steam

The energy required for the phase transition using the latent heat of vaporization \(L = 2.26 \times 10^6 \, \mathrm{J/kg}\) is:\[Q_2 = mL = m \times 2.26 \times 10^6\]

Step 5: Solve for the Mass of Water Boiled Away

Total energy \(E_{boil}\) is used to first increase the temperature to boiling point and then to convert water into steam. The sum is given by:\[E_{boil} = Q_1 + Q_2 = 4186 \times 90 \times m + 2.26 \times 10^6 \times m\]Substituting for \(E_{boil}\):\[1.33 \times 10^{23} = (376740 + 2.26 \times 10^6) \times m\]Solve for \( m \):\[m \approx \frac{1.33 \times 10^{23}}{2.63674 \times 10^6}\]\[m \approx 5.04 \times 10^{16} \mathrm{kg}\]

Step 6: Compare with Lake Superior's Mass

The boiled water mass \(5.04 \times 10^{16}\, \mathrm{kg}\) is much larger than Lake Superior's mass \(\approx 2 \times 10^{15}\,\mathrm{kg}\), showing the significant impact of the asteroid.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It's an essential concept in understanding the impact forces in astrodynamics, especially when celestial bodies like asteroids move at high speeds. The kinetic energy (\( KE \)) of an object can be calculated using the formula:

• \[ KE = \frac{1}{2}mv^2 \]

Here, \( m \) represents the object's mass and \( v \) its velocity. In the case of the asteroid impacting Earth, its enormous speed and mass result in a colossal amount of kinetic energy. This energy gets transferred to the environment, and partly contributes to phenomena like warming water or even vaporizing it upon collision.
When considering scenarios such as the collision of an asteroid with Earth, understanding kinetic energy offers insights into potential effects and hazards, like massive energy transfer to Earth's surface, posing significant impacts on the environment and life.

Specific Heat Capacity

Specific heat capacity is a property of a material that denotes the amount of heat required to change its temperature by a given amount. It's a fundamental concept in physics and chemistry, showing how materials respond to heat energy. The specific heat capacity (\( c \)) of water is 4186 \( \mathrm{J/kg \, ^\circ C} \). This means that each kilogram of water requires 4186 joules of energy to increase its temperature by one degree Celsius.
When calculating the energy needed to heat water, you use the formula:

• \[ Q = mc\Delta T \]

In this formula, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. This calculation is crucial for determining how much energy from the asteroid's impact contributes to raising the water's temperature before it reaches boiling. The specific heat capacity of water is relatively high, indicating that water can absorb large amounts of heat without significantly changing its temperature.

Phase Change

The concept of phase change involves the transition of substances between different states of matter: solid, liquid, and gas. During a phase change, such as from liquid water to steam (gas), the substance absorbs or releases a specific amount of energy without affecting its temperature. This energy input or release is critical during processes like boiling, as it characterizes the transition energy needed for state change.
Phase changes are inherently related to physical conditions like pressure and temperature. In a scenario where an asteroid impacts the Earth, a huge amount of energy can cause a phase change in water, converting it to steam after reaching the boiling point. This energy, known as latent heat, does not raise the temperature but supplies the necessary energy to break the bonds that hold water molecules in the liquid state, allowing them to transition into a gaseous form.

Latent Heat of Vaporization

The latent heat of vaporization is the amount of heat a substance requires to change from a liquid to a gas without changing its temperature. Water's latent heat of vaporization is approximately 2.26 \( \times 10^6 \) \( \mathrm{J/kg} \). This value signifies the robust energy needed during the phase transition of water into steam.
In real-world events like an asteroid impact, a part of the kinetic energy gets converted into thermal energy, which heats the water. After the water reaches its boiling point, any additional energy contributes to the phase change, quantified by the latent heat of vaporization. This concept helps to calculate how much of the asteroid's energy would be used to turn liquid water into steam in practical scenarios.

• The formula to determine the energy required for vaporization is:\[ Q = mL \]
• Here, \( m \) is the mass of the substance, and \( L \) is the latent heat of vaporization.

This calculation is critical when predicting environmental impacts, such as how much water might transition to steam in an extreme event like an asteroid's ocean landing.