Question

An asteroid with a diameter of 10 km and a mass of \(2.60 \times 10{^1}{^5} kg\) impacts the earth at a speed of 32.0 km/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\(^\circ\)C), what mass of water will be boiled away by the collision? (For comparison, the mass of water contained in Lake Superior is about \(2 \times 10{^1}{^5} kg\).)

Short answer

Approximately 5050 kg of water will be boiled away by the collision.

Step-by-step solution

Calculate the Kinetic Energy of the Asteroid

The kinetic energy (KE) of an object can be calculated using the formula: \[ KE = \frac{1}{2} m v^2 \]where \( m = 2.60 \times 10^5 \text{ kg} \) is the mass of the asteroid and \( v = 32.0 \times 10^3 \text{ m/s} \) is the speed. So, \[ KE = \frac{1}{2} \times 2.60 \times 10^5 \times (32.0 \times 10^3)^2 \text{ Joules} \].Calculating this gives:\[ KE \approx 1.3312 \times 10^{15} \text{ Joules} \].

Determine Energy Used to Boil Water

Only 1.00% of the asteroid's kinetic energy is used to boil water. Thus, the energy used to boil water, \( E_{boil} \), is:\[ E_{boil} = 0.01 \times 1.3312 \times 10^{15} \text{ Joules} \].Calculating this gives:\[ E_{boil} \approx 1.3312 \times 10^{13} \text{ Joules} \].

Calculate the Energy Required to Boil 1 kg of Water

To boil water from an initial temperature \( T_i = 10.0^{\circ}C \) to \( 100^{\circ}C \) and then vaporize it, the total energy required for 1 kg is \[ E = m c \Delta T + m L_v \] where \( c = 4.186 \text{ kJ/kg°C} \) is the specific heat, \( L_v = 2260 \text{ kJ/kg} \) is the latent heat of vaporization, and \( \Delta T = 90^{\circ}C \).So for 1 kg, \[ E = 1 \times 4.186 \times 90 + 1 \times 2260 \text{ (kJ)} \] \[ E = 376.74 + 2260 \] \[ E = 2636.74 \text{ kJ = 2.63674 \times 10^6 J} \].

Find the Mass of Water Boiled Away

To find the total mass \( m_{water} \) of water that can be boiled away, we use the energy relation:\[ m_{water} = \frac{E_{boil}}{E_{1}} \]where \( E_{1} = 2.63674 \times 10^6 \text{ Joules} \) is the energy required to boil 1 kg of water. Thus, \[ m_{water} = \frac{1.3312 \times 10^{13}}{2.63674 \times 10^6} \text{ kg} \].Calculating this gives:\[ m_{water} \approx 5050 \text{ kg} \].

Key concepts

Asteroid Impact

When an asteroid crashes into Earth, it releases an enormous amount of energy. This energy stems from its mass and the speed at which it is traveling. The concept is similar to how a moving car has more energy the heavier and faster it moves. Imagine a giant space rock with tremendous speed; the impact creates a violent explosion. This explosion can dramatically affect our planet, causing everything from massive craters to climate changes. In this case, the asteroid hits the Pacific Ocean, creating an impact that involves converting the asteroid’s kinetic energy into thermal energy, which subsequently heats and boils a significant mass of ocean water. Understanding asteroid impacts helps scientists assess potential risks from future encounters with such celestial objects.

Energy Conversion

Energy conversion is a fundamental principle where energy changes from one form to another. During the asteroid impact, the primary conversion occurs from kinetic energy (energy due to motion) to thermal energy (heat). This transformation process is crucial because it demonstrates how energy is conserved but changes its form to create different effects. In our case, a small fraction (1%) of the kinetic energy of the asteroid is converted into heat. This heat is what boils the ocean water. Energy never just disappears; it morphs into other types of energy while performing work on its surroundings. The conversion from kinetic to thermal energy explains how mechanical impacts can generate heat capable of causing water to vaporize and affect the environment.

• Kinetic Energy: The energy an object possesses by being in motion.
• Thermal Energy: The internal energy from heat, which can change the state of matter, such as boiling water.

Latent Heat of Vaporization

Latent heat of vaporization is the amount of energy required to turn a unit mass of a liquid into vapor without altering its temperature. In simpler terms, it’s the heat you need to add to a liquid for it to evaporate completely. For water, this value is significantly substantive; it takes a large amount of energy to convert water into steam compared to just heating it. During an asteroid impact, when water is boiled and vaporized, this concept plays a critical role in calculations. Once water reaches 100°C, heat continues to flow into it but instead of increasing temperature, it changes state from liquid to gas. This energy requirement is quantified as the latent heat of vaporization, which for water is approximately 2260 kJ/kg. Understanding this concept is key in predicting the environmental effects and energy distribution following high-impact events like asteroid collisions.

Specific Heat Capacity

Specific heat capacity refers to the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. It might sound technical, but consider it the resistance of a material to temperature change. Water has a high specific heat capacity, meaning that it takes a lot of energy to change its temperature compared to other substances. This property becomes relevant when calculating the amount of water that can be heated by the asteroid's impact. Before vaporization, the water must first be heated to boiling point from an initial temperature of 10°C. The specific heat capacity of water, which is about 4.186 kJ/kg°C, tells us how much energy this heating process requires. Understanding specific heat capacity helps students and scientists determine how substances react to energy inputs and affects energy management in environmental scenarios.