Question

In the spirit of outlandish extrapolations, if we carry forward a \(2.3 \%\) growth rate \((10 \times\) per century \()\), how long would it take to go from our current \(18 \mathrm{TW}\left(18 \times 10^{12} \mathrm{~W}\right)\) consumption to annihilating an entire earth-mass planet every year, converting its mass into pure energy using \(E=m c^{2} ?\) Things to know: Earth's mass is \(6 \times 10^{24} \mathrm{~kg} ; c=3 \times 10^{8} \mathrm{~m} / \mathrm{s} ;\) the result is in Joules, and one Watt is one Joule per second.

Short answer

Answer: To find the answer, please follow the step-by-step solution provided and calculate the value of 'x' in the final equation. This value will represent the number of centuries it would take to reach the required energy consumption level.

Step-by-step solution

Calculate the energy released by converting Earth's mass into pure energy

Using the equation E=mc^2, we can calculate the energy that would be released by converting Earth's mass into pure energy. Earth's mass is 6 x 10^24 kg, and the speed of light c is 3 x 10^8 m/s: \( E = m \cdot c^2 \) Plug in the values: \[E = 6 \times 10^{24} kg \times (3 \times 10^8ms^{-1})^2\]

Calculate the energy released per second

To find out the energy conversion rate (in watts), we need to convert the energy released per year to energy released per second. There are 365 days in a year, 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute: Energy per second = \(\frac{Energy}{365 \cdot 24 \cdot 60 \cdot 60}\)

Calculate the time it would take for the consumption level to reach this energy conversion rate

We are given a growth rate of 2.3% (0.023), which means the energy consumption level grows 10 times per century. Let's assume that the time it takes is x centuries. We can set up the equation: Current consumption level * (1 + growth rate)^(10 * x) = Energy consumption level needed Plug in the current consumption level (18x10^12 W) and the energy consumption level needed (from step 2): \(18 \times 10^{12} W \times (1 + 0.023)^{10 \times x} = Energy\_Per\_Second\)

Solve for x

Rearrange the equation to isolate x: \(x = \frac{\log{(\frac{Energy\_Per\_Second}{18 \times 10^{12}W })} - \log{(1+0.023)}}{10 \cdot \log{(1+0.023)}}\) Calculate the value of x, and that will be the number of centuries it would take for the consumption level to reach the energy conversion rate needed to annihilate an entire earth-mass planet every year.

Key concepts

Energy Consumption Growth

Understanding the trajectory of energy consumption growth is critical in anticipating future energy needs and environmental impacts. The concept of energy consumption growth is tied to how the demand for energy has been increasing due to the expansion of global economies, technological advancements, and population growth. When we say there's a growth rate of 2.3% in energy consumption, it means that each year, the amount of energy used is 102.3% of what was used the previous year.

This seemingly small percentage can compound rapidly due to the nature of exponential growth, leading to a considerable increase in energy demand over a relatively short period. For instance, a growth rate of 2.3% per year translates to a doubling of energy consumption roughly every 31 years. Therefore, extrapolating this rate of growth can help us project future energy requirements, albeit with a requirement for critical analysis since physical limits and sustainability concerns may intervene to slow or alter this growth trajectory.

Mass-Energy Equivalence

Mass-energy equivalence is epitomized by Albert Einstein's famous equation, \( E=mc^2 \), which states that energy (\(E\)) and mass (\(m\)) are interchangeable; they are different forms of the same thing. In this equation, \(c\) represents the speed of light in a vacuum, which is approximately \(3 \times 10^8\) meters per second.

The equation carries profound implications, one being that a small amount of mass can be converted into a tremendous amount of energy, illustrated by the immense energy release in nuclear reactions. It's this principle that allows us to theoretically consider the energy potential of annihilating an earth-mass planet. Although purely hypothetical and not feasible with current technology, this exercise demonstrates the vast quantities of energy tied up in even seemingly small amounts of mass.

Exponential Growth Calculations

Exponential growth calculations are crucial for modeling processes that grow at a rate proportional to their current value. This type of growth is commonly described by an exponential function, most easily understood through a constant percentage rate over fixed time intervals.

For example, if energy consumption grows at a constant rate of 2.3% annually, we can calculate how consumption will increase over time using the formula \( P = P_0 \times (1 + r)^{nt} \), where \( P \) is the final amount of energy, \( P_0 \) is the initial amount, \( r \) is the growth rate, \( n \) is the number of times the growth is compounded per unit time, and \( t \) is the time in those units. Through these calculations, we can predict future energy needs, providing valuable foresight into our potential resource requirements and allowing us to prepare accordingly.

Sustainable Energy

Sustainable energy refers to energy sources and systems that provide for the current needs without compromising the ability of future generations to meet their energy needs. This includes renewable resources like solar, wind, hydro, and geothermal power. Sustainable energy prioritizes the reduction of greenhouse gas emissions, aims to diminish the reliance on fossil fuels, and considers the environmental and economic impact of energy production and consumption.

Sustainable energy is fundamentally important as it offers a pathway to mitigate climate change and reduce environmental degradation. With the ever-growing energy consumption rates, transitioning to sustainable energy sources is not just an environmental imperative but also essential for long-term economic and social stability. While the exercise of annihilating an entire planet for energy is purely theoretical and not consistent with sustainable practices, it underscores the immense energy challenges we face and the importance of developing sustainable solutions to meet our energy demands.