Question

Consider a comet about 2 kilometres across with a mass of \(4 \times 10^{12}\) kg. Assume that it crashes into Earth at a speed of 30,000 meters per second (about 67,000 miles per hour) .a. What is the total energy of the impact, in joules? (Hint: The "kinetic energy" formula tells us that the impact energy in joules will be \(\frac{1}{2} \times m \times v^{2}\), where \(m\) is the comet's mass in kilograms and \(v\) is its speed in meters per second.) b. A 1-megaton nuclear explosion releases about \(4 \times 10^{15}\) joules of energy. How many such nuclear bombs would it take to release as much energy as the comet impact? c. Based on your answers, comment on the degree of devastation the comet might cause.

Short answer

The comet impact releases \(1.8 \times 10^{21}\) joules, equivalent to 450,000 nuclear bombs, indicating massive devastation.

Step-by-step solution

Understanding Kinetic Energy Formula

To find the impact energy in joules, use the formula for kinetic energy: \( KE = \frac{1}{2} \times m \times v^2 \), where \( m \) is mass and \( v \) is velocity.

Input Values into the Formula

Plug in the given values into the kinetic energy formula: \( m = 4 \times 10^{12} \text{ kg} \) and \( v = 30,000 \text{ m/s} \). The equation becomes \( KE = \frac{1}{2} \times 4 \times 10^{12} \times (30,000)^2 \).

Calculate the Energy

Calculate the kinetic energy: \( KE = \frac{1}{2} \times 4 \times 10^{12} \times 9 \times 10^{8} = 18 \times 10^{20}\). Simplifying further gives \( KE = 1.8 \times 10^{21} \text{ Joules} \).

Compare Energy to Nuclear Explosion

A 1-megaton nuclear explosion releases \( 4 \times 10^{15} \text{ Joules} \). Divide the impact energy by the energy of one explosion: \( \frac{1.8 \times 10^{21}}{4 \times 10^{15}} \approx 4.5 \times 10^{5} \) nuclear bombs.

Analyze the Degree of Devastation

Given that the impact energy is equivalent to about 450,000 nuclear bombs, the comet impact would result in massive devastation, potentially leading to global climate changes similar to those theorized for the extinction of the dinosaurs.

Key concepts

Impact Energy

The term "impact energy" refers to the kinetic energy that a moving object, like a comet, transfers upon colliding with another body, such as Earth. Kinetic energy is a form of energy associated with motion. To determine the impact energy of our comet, we use the kinetic energy formula:

\[ KE = \frac{1}{2} \times m \times v^2 \]
In this equation, \( m \) represents mass and \( v \) represents velocity. By plugging the values given in the exercise like mass \( m = 4 \times 10^{12} \) kg and velocity \( v = 30,000 \) m/s into the formula, the impact energy is calculated to be \( 1.8 \times 10^{21} \) Joules.

This immense amount of energy demonstrates the destructive potential of swift-moving celestial bodies.

Nuclear Explosion Energy Comparison

To appreciate the might of a comet impact, it's helpful to compare it against something more familiar, like a nuclear explosion. A single 1-megaton nuclear bomb is incredibly powerful, releasing approximately \( 4 \times 10^{15} \) Joules of energy.

However, when we compare this to our comet impact, which releases about \( 1.8 \times 10^{21} \) Joules, the difference is staggering.

• The comet impact equates to about 450,000 nuclear bombs.
• This comparison highlights the comet's potential for widespread destruction, well beyond human-made disasters.

Understanding this scale helps to grasp how an impact event can be catastrophic on a global level, affecting climates, landscapes, and life on Earth.

Comet Impact Devastation

A comet impact is one of the few natural events capable of causing mass extinction due to its colossal energy. Here's what can happen during and after such an impact:

• **Immediate Aftermath:** The initial strike can create a massive crater and instantly vaporize everything at the impact site. The energy release is equivalent to hundreds of thousands of nuclear explosions.
• **Atmospheric Effects:** Dust and debris are thrown into the atmosphere, potentially blocking out sunlight and lowering global temperatures. This is often referred to as an "impact winter."
• **Environmental Shifts:** The altered climate can lead to widespread crop failures, disruptions in ecosystems, and possibly a collapse in food chains.

Consequently, an impact of this magnitude has the potential to reshape Earth's environment and life, just like the event believed to have led to the extinction of the dinosaurs.

Energy Calculation Formula

Calculating energy, particularly kinetic energy for large objects like comets, involves understanding how mass and velocity contribute to an object's overall energy. Let's dive into the formula a bit more:

\[ KE = \frac{1}{2} \times m \times v^2 \]

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

The formula tells us that energy grows quadratically with velocity, meaning doubling the speed results in a fourfold increase in energy. This is why high-speed impacts are so devastating—although the mass might not be excessively huge, the velocity amplifies energy greatly.

Plugging values into the formula is straightforward, but it's crucial to handle units with care to ensure accurate results. Understanding stages of calculation helps in tackling such physics problems effectively.