Question

Be sure to show all calculations clearly and state your final answers in complete sentences. The Multistage Rocket Equation. The rocket equation takes a slightly different form for a multistage rocket: \\[ v=n v_{\mathrm{e}} \ln \left(\frac{M_{\mathrm{i}}}{M_{\mathrm{f}}}\right) \\] where \(n\) is the number of stages. a. Suppose a rocket has three stages with mass ratio \(M_{\mathrm{i}} / M_{\mathrm{f}}=3.4\) and engines that produce an exhaust velocity of \(3 \mathrm{km} / \mathrm{s}\) What is its final velocity? Is it sufficient to escape Earth? b. Suppose a rocket has 100 stages with mass ratio \(M_{\mathrm{i}} / M_{\mathrm{f}}=3.4\) and engines that produce an exhaust velocity of \(3 \mathrm{km} / \mathrm{s}\) What is its final velocity? Compare it to the speed of light.

Short answer

Three-stage rocket: 11,014.2 m/s, insufficient to escape Earth. 100-stage rocket: 367,140 m/s, much slower than light speed.

Step-by-step solution

Understand the Rocket Equation

The multistage rocket equation is \( v = n v_{\mathrm{e}} \ln \left(\frac{M_{\mathrm{i}}}{M_{\mathrm{f}}}\right) \), where \( v \) is the final velocity, \( n \) is the number of stages, \( v_{\mathrm{e}} \) is the exhaust velocity, and \( \frac{M_{\mathrm{i}}}{M_{\mathrm{f}}} \) is the mass ratio. We will use this formula to calculate the final velocity for each scenario.

Calculate Final Velocity for Three Stages

For a three-stage rocket with \( n = 3 \), \( v_{\mathrm{e}} = 3000 \) m/s, and \( \frac{M_{\mathrm{i}}}{M_{\mathrm{f}}} = 3.4 \), the final velocity is calculated as follows:\[ v = 3 \times 3000 \times \ln(3.4) \]Calculate \( \ln(3.4) \) and multiply by the other terms:\[ \ln(3.4) \approx 1.2238 \]\[ v = 3 \times 3000 \times 1.2238 = 11014.2 \text{ m/s} \]

Assess if 11,014.2 m/s is Sufficient to Escape Earth

The escape velocity from Earth is approximately \( 11,186 \) m/s. The calculated velocity for a three-stage rocket is \( 11,014.2 \) m/s, which is slightly below the escape velocity, indicating it is insufficient to escape Earth's gravitational pull.

Calculate Final Velocity for 100 Stages

For a 100-stage rocket, using the same exhaust velocity and mass ratio:\[ v = 100 \times 3000 \times \ln(3.4) \]Using the previously calculated \( \ln(3.4) \approx 1.2238 \):\[ v = 100 \times 3000 \times 1.2238 = 367,140 \text{ m/s} \]

Compare 367,140 m/s to the Speed of Light

The speed of light is approximately \( 299,792,458 \) m/s. The calculated velocity for a 100-stage rocket is \( 367,140 \) m/s, which is significantly slower than the speed of light.

Key concepts

Rocket Stages

In the world of rockets, stages are like the different chapters of a journey. Each stage serves a specific purpose and is equipped with its own engines and fuel. As the rocket ascends into space, it jettisons each stage when its fuel is depleted, shedding heavy weight and enabling the subsequent stage to take over. This design improves efficiency by allowing the remaining stages to achieve higher speeds, without the burden of unnecessary mass from spent fuel tanks and engines.
The idea of multistage rockets is crucial because it allows for greater acceleration by optimizing the resources available at different phases of a rocket's flight. By breaking the journey into stages, engineers maximize the velocity achievable from a given amount of fuel. Like climbing a mountain in a relay race, each stage plays a part in pushing the rocket towards its final destination.

Mass Ratio

The mass ratio in rocket science is a measure of efficiency. It describes the proportion between the initial total mass of the rocket (including fuel) and the final mass (after the fuel has been spent). This ratio is crucial for understanding how effectively a rocket can leverage its fuel supply.
A higher mass ratio signifies that a significant portion of the rocket's original mass was fuel, enabling it to achieve greater changes in velocity. For example, in our exercise, a mass ratio of 3.4 means the rocket started with 3.4 times more mass than it would have without any fuel. Understanding the mass ratio helps in determining how fast the rocket can go and how much of its mass it needs to shed to reach the desired speed.

• The formula for the mass ratio is \( \frac{M_{\mathrm{i}}}{M_{\mathrm{f}}} \), where \( M_{\mathrm{i}} \) is the initial mass and \( M_{\mathrm{f}} \) is the final mass.
• This ratio is crucial for calculating the final velocity using the rocket equation.

Exhaust Velocity

Exhaust velocity is a critical factor impacting how fast a rocket can travel. It refers to the speed at which exhaust gases are expelled from the rocket's engines. The higher the exhaust velocity, the faster these gases shoot out, pushing the rocket forward with greater force. Think of it like a balloon zooming across the room when you let go—it moves because air is rapidly escaping.
In our exercise, the exhaust velocity is given as 3 km/s (or 3000 m/s). This metric is central to the rocket equation, playing a direct part in determining the rocket's final velocity. The larger the exhaust velocity, the more effectively the rocket can leverage its mass ratio to achieve higher speeds. Calculating the exhaust velocity accurately allows engineers to predict the performance and capabilities of a rocket, ensuring it meets its mission criteria.

Escape Velocity

Escape velocity is the speed a rocket needs to break free from a celestial body's gravitational pull, without further propulsion. On Earth, this velocity is about 11,186 m/s. It represents the balance point between a rocket's momentum and Earth's gravitational force.
For our three-stage rocket with a final velocity of 11,014.2 m/s, the velocity is just shy of reaching this critical speed, meaning the rocket would not be able to leave Earth's grasp unless aided by additional propulsion methods. However, these calculations demonstrate how one might conceive booster strategies to cross that threshold. The concept of escape velocity is fundamental for missions aiming to reach other planets or moons.

Speed of Light Comparison

In the realm of physics, comparing a rocket's velocity to the speed of light illustrates the enormous speed gap. The speed of light in a vacuum is about 299,792,458 m/s. Even our 100-stage rocket, reaching 367,140 m/s, pales in comparison. This intriguing comparison highlights the challenges of high-speed travel and underscores the superior speed of electromagnetic forces compared to chemical propulsion.
Although reaching the speed of light with current technology remains within the realm of science fiction, comparing rocket velocities serves as a valuable tool for students to grasp the scale of physical constants. It emphasizes the need for advancing propulsion technology if humanity is ever to reach interstellar speeds.