Question

One important characteristic of rocket engines is the specific impulse, which is defined as the total impulse (time integral of the thrust) per unit ground weight of fuel/oxidizer expended. (The use of weight, instead of mass, in this definition is due to purely historical reasons.) a) Consider a rocket engine operating in free space with an exhaust nozzle speed of \(v\). Calculate the specific impulse of this engine. b) A model rocket engine has a typical exhaust speed of \(v_{\text {toy }}=800 . \mathrm{m} / \mathrm{s}\). The best chemical rocket engines have exhaust speeds of approximately \(v_{\text {chem }}=4.00 \mathrm{~km} / \mathrm{s} .\) Evaluate and compare the specific impulse values for these engines.

Short answer

Question: Calculate the specific impulse for a rocket engine in free space with an exhaust nozzle speed of \(v\) and compare the specific impulse values for a toy rocket engine with an exhaust speed of \(800 \mathrm{~m/s}\) and a chemical rocket engine with an exhaust speed of \(4.00 \mathrm{~km/s}\). Answer: The specific impulse for a rocket engine in free space with an exhaust nozzle speed of \(v\) is \(I_{sp}(v) = \frac{v}{9.81\ \mathrm{m/s^2}}\). The specific impulse for the toy rocket engine is approximately \(81.55\), while the specific impulse for the chemical rocket engine is approximately \(407.95\). The chemical rocket engine has a significantly higher specific impulse, making it more efficient in terms of thrust generated per unit weight of the fuel used.

Step-by-step solution

Write down the formula for specific impulse

We'll use the formula \(I_{sp} = \frac{v_{e}}{g_0}\), where \(I_{sp}\) is the specific impulse, \(v_{e}\) is the exhaust velocity, and \(g_0\) is the standard acceleration due to gravity.

Calculate the specific impulse

In this case, the exhaust nozzle speed of the engine is \(v\). So, \(v_{e} = v\). Using the formula, we get \(I_{sp} = \frac{v}{g_0}\). Since we are looking for the specific impulse in terms of \(v\), the solution is \(I_{sp}(v) = \frac{v}{9.81\ \mathrm{m/s^2}}\). #b) Evaluate and compare the specific impulse values for the toy rocket engine and the chemical rocket engine#

Calculate the specific impulse for the toy rocket engine

For the toy rocket engine, the exhaust speed is given as \(v_{\text {toy }}=800 \mathrm{~m/s}\). Using the specific impulse formula and plugging in the values, we get \(I_{sp_{\text{toy}}} = \frac{800\ \mathrm{m/s}}{9.81\ \mathrm{m/s^2}} \approx 81.55\).

Calculate the specific impulse for the chemical rocket engine

For the chemical rocket engine, the exhaust speed is given as \(v_{\text {chem }}=4.00 \mathrm{~km/s}\). We need to convert this to meters per second: \(v_{\text {chem }}=4000 \mathrm{~m/s}\). Using the specific impulse formula and plugging in the values, we get \(I_{sp_{\text{chem}}} = \frac{4000\ \mathrm{m/s}}{9.81\ \mathrm{m/s^2}} \approx 407.95\).

Compare the specific impulse values of the toy and chemical rocket engines

Comparing the specific impulse values, we can see that the chemical rocket engine has a significantly higher specific impulse (\(407.95\)) compared to the toy rocket engine (\(81.55\)). This means that the chemical rocket engine is more efficient in terms of thrust generated per unit weight of the fuel used.

Key concepts

Rocket Engines

Rocket engines are marvels of engineering designed to propel objects through space. They operate on the principle of action and reaction, as stated by Newton's third law of motion: for every action, there is an equal and opposite reaction. When a rocket engine expels exhaust gases at high speed in one direction, it generates thrust that propels the rocket forward in the opposite direction. This thrust is what allows rockets to launch from Earth, maneuver in space, and even land on other celestial bodies.

Unlike vehicles on Earth, which use air to create lift or traction, rockets must carry both their fuel and oxidizer, allowing them to operate in the vacuum of space. The efficiency and effectiveness of a rocket are highly dependent on its engine design, materials used, and the type of fuel and oxidizer. From the iconic engines that powered the Apollo missions to the reusable Merlin engines on the Falcon 9 rockets, the evolution of rocket engines continues to expand our reach into the universe.

Exhaust Velocity

Exhaust velocity is a crucial parameter in the context of rocket engines. It refers to the speed at which exhaust gases are ejected from the engine. The higher the exhaust velocity, the more effective the engine is at converting the rocket's fuel into usable thrust. In a simplified explanation, a rocket burns its propellant to produce hot gases. These gases are then expelled through a nozzle to create forward momentum.

Exhaust velocity not only influences the thrust, but also the efficiency of a rocket. A higher exhaust velocity results in a better specific impulse, indicating a more efficient engine. Engineers tirelessly work on improving the exhaust velocity of rocket engines through advanced designs and materials, understanding that even small increases can have substantial effects on the rocket's performance and the payload it can deliver to space.

Thrust

Thrust is the force generated by a rocket engine to overcome the gravitational pull of the Earth and other forces such as air resistance. It's the engine's 'push' that lifts a rocket from the ground into space. The magnitude of the thrust dictates how much weight the rocket can carry, which includes the payload, fuel, and the rocket structure itself.

In the context of the exercise, the specific impulse provides a way to measure how efficiently a rocket engine generates thrust. A higher specific impulse indicates that a rocket can produce more thrust for a given amount of propellant mass. Achieving an optimal level of thrust while managing the weight of the propellant is one of the key challenges in rocket design. Engineers must consider these factors when developing both small and large-scale rocket engines for varying mission requirements.

Standard Acceleration due to Gravity

The standard acceleration due to gravity, often denoted as \(g_0\), is a constant value used in physics to represent the nominal gravitational acceleration on Earth's surface. Its average value is about \(9.81 \mathrm{m/s^2}\). This constant is important in rocketry calculations, especially when determining the specific impulse of rocket engines.

In specific impulse calculations, as shown in the exercise, \(g_0\) serves as a reference point to normalize the thrust force due to the varying gravity encountered during spaceflight. By using \(g_0\) as the denominator in the formula for specific impulse, we obtain a value that can be compared consistently across different rocket engines, regardless of the variations in gravitational force at different altitudes and celestial bodies. Understanding \(g_0\) is essential for students trying to grasp the complexity of rocket dynamics and the impact of Earth's gravity on space travel.