Question

Be sure to show all calculations clearly and state your final answers in complete sentences. Cruise Ship Energy. Suppose we have a spaceship about the size of a typical ocean cruise ship today, which means it has a mass of about 100 million kilograms, and we want to accelerate the ship to a speed of \(10 \%\) of the speed of light. a. How much energy would be required? (Hint: You can find the answer simply by calculating the kinetic energy of the ship when it reaches its cruising speed; because \(10 \%\) of the speed of light is still small compared to the speed of light, you can use the formula that tells us that kinetic energy \(=\frac{1}{2} \times m \times v^{2} .\) ) b. How does your answer compare to total world energy use at present, which is about \(5 \times 10^{20}\) joules per year? c. Suppose the cost of energy is \(3 \varphi\) per 1 million joules. Using this price, how much would it cost to generate the energy needed by this spaceship?

Short answer

The energy required is \(4.5 \times 10^{22}\) joules, 90 times the current world energy use. The cost is \(1.35 \times 10^{17} \varphi\).

Step-by-step solution

Understanding the Problem

We need to find the kinetic energy required to accelerate a 100 million-kilogram spaceship to 10% of the speed of light. The speed of light is approximately \( c = 3 \times 10^8 \) meters per second.

Writing the Kinetic Energy Formula

Kinetic energy \( (KE) \) is given by the formula \( KE = \frac{1}{2} \times m \times v^2 \), where \( m \) is mass and \( v \) is velocity. Here, \( m = 100 \times 10^6 \) kg and \( v = 0.1c = 0.1 \times 3 \times 10^8 = 3 \times 10^7 \) m/s.

Calculating the Kinetic Energy

Substitute the values into the kinetic energy formula: \[ KE = \frac{1}{2} \times 100 \times 10^6 \times (3 \times 10^7)^2 \] First, calculate \((3 \times 10^7)^2 = 9 \times 10^{14}\). Next, calculate:\[ KE = \frac{1}{2} \times 100 \times 10^6 \times 9 \times 10^{14} = 4.5 \times 10^{22} \text{ joules} \]

Comparing to World Energy Use

The world energy consumption is \(5 \times 10^{20}\) joules per year. Comparing this:\[ \frac{4.5 \times 10^{22}}{5 \times 10^{20}} = 90 \] The energy required is 90 times the annual world energy consumption.

Calculating the Cost

Energy cost given is \(3 \varphi\) per 1 million joules. Convert the energy required:\[ 4.5 \times 10^{22} \text{ joules} = 4.5 \times 10^{16} \text{ million joules} \]Hence, the cost is: \[ 4.5 \times 10^{16} \times 3 \varphi = 1.35 \times 10^{17} \varphi \]

Key concepts

Energy Consumption of Spaceships

Understanding the energy consumption needed to propel a spaceship, like one that weighs around 100 million kilograms, to speeds comparable to fractions of the speed of light can be quite intriguing. To determine this, we use the concept of kinetic energy, which is the energy an object possesses due to its motion. Kinetic energy is calculated using the formula: \[ KE = \frac{1}{2} \times m \times v^2 \] Where\( m \) stands for mass and \( v \) is the velocity of the object. When our spaceship accelerates to reach 10% of the speed of light, calculated at approximately \( 3 \times 10^7 \) meters per second (as speed of light \( c \) is \( 3 \times 10^8 \) m/s), its kinetic energy amounts to \( 4.5 \times 10^{22} \) joules! This staggering amount of energy accounts for 90 times the total energy consumption globally each year. Such monumental energy requirements highlight the challenges and magnitude of energy needs when considering space exploration at relativistic speeds.

Exploring the Speed of Light

The speed of light, symbolized as \( c \), is one of the fundamental constants in physics and holds a value of approximately \( 3 \times 10^8 \) meters per second. It's incredibly fast, and nothing known can travel faster. The concept and value of the speed of light are crucial in the realm of physics because it not only sets a universal speed limit but also plays a vital role in theories of relativity and influences our understanding of time and space. When considering velocities such as 10% of the speed of light, we enter realms where traditional Newtonian physics gives way to Einstein's theory of relativity. However, at speeds much less than \( c \), like our spaceship's velocity \( 0.1c \), traditional formulas for kinetic energy, such as \( KE = \frac{1}{2} m v^2 \), provide a sufficiently accurate estimate of the energy required. Preparing space vessels to approach such high-speed journeys is a topic of intense interest and challenges the limits of current energy technologies.

Impact of Spaceship Mass

The mass of a spaceship, especially one akin to the size of a modern ocean cruise liner with a mass of approximately 100 million kilograms, is a critical factor when calculating energy consumption for space travel. This mass directly feeds into the kinetic energy formula, emphasizing how even relatively small changes in mass can lead to tremendous differences in energy needs when accelerating to high velocities.

• Mass is one of the key contributors to the ship's momentum and inertia.
• Heavier ships require more energy to achieve the same change in speed compared to lighter ships.
• The substantial mass of the spaceship requires careful consideration of the fuels and propulsion systems needed to achieve desired speed efficiently and sustainably.

Understanding and manipulating this aspect of space travel remains a primary challenge for aerospace engineers and scientists aiming to balance safety, efficiency, and capability. Each kilogram counts significantly in the world of space engineering, where achieving optimal energy use without compromising safety is paramount.