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: Compare the specific impulse values of a model rocket engine and the best chemical rocket engines. Answer: The specific impulse value for the model rocket engine is 81.55 seconds, while the specific impulse value for the best chemical rocket engines is 407.75 seconds. The best chemical rocket engines are much more efficient in utilizing their fuel/oxidizer per unit of ground weight, compared to the model rocket engines.

Step-by-step solution

Write down the formula for specific impulse

The specific impulse (Isp) can be calculated using the formula: \(Isp = \frac{Total\ Impulse}{Ground\ Weight\ of\ Fuel/Oxidizer\ Expended}\) In free space, the total impulse can be represented as the product of the mass flow rate (\(\dot{m}\)) of the exhaust, exhaust speed (\(v\)), and burn time (\(t\)). The Ground Weight of Fuel/Oxidizer Expended is given by the product of the mass of the fuel/oxidizer (\(m\)) and the acceleration due to gravity (\(g\)). So, the formula can be rewritten as: \(Isp = \frac{\dot{m} \cdot v \cdot t}{m \cdot g}\)

Simplify the formula

The mass flow rate (\(\dot{m}\)) multiplied by the burn time (\(t\)) will result in the mass of the fuel/oxidizer expended (\(m\)). Therefore, the formula can be further simplified as: \(Isp = \frac{v}{g}\) Now, we can use the simplified formula to calculate the specific impulse of the engine. #b) Evaluate and compare the specific impulse values for a model rocket engine and the best chemical rocket engines#

Calculate specific impulse for the model rocket engine

We are given the exhaust speed for the model rocket engine (\(v_{\text {toy }}\)) as 800 m/s. The acceleration due to gravity (\(g\)) is approximately 9.81 m/s². To calculate the specific impulse for the model rocket engine, use the simplified formula: \(Isp_{\text{toy}} = \frac{v_{\text{toy}}}{g}\) \(Isp_{\text{toy}} = \frac{800\,\text{m/s}}{9.81\,\text{m/s}^2}\) \(Isp_{\text{toy}} \approx 81.55\,\text{s}\)

Calculate specific impulse for the best chemical rocket engines

We are given the exhaust speed for the best chemical rocket engines (\(v_{\text {chem }}\)) as 4.00 km/s, which is equal to 4000 m/s. The acceleration due to gravity (\(g\)) is approximately 9.81 m/s². To calculate the specific impulse for the best chemical rocket engines, use the simplified formula: \(Isp_{\text{chem}} = \frac{v_{\text{chem}}}{g}\) \(Isp_{\text{chem}} = \frac{4000\,\text{m/s}}{9.81\,\text{m/s}^2}\) \(Isp_{\text{chem}} \approx 407.75\,\text{s}\)

Compare the specific impulse values

Now that we have calculated the specific impulse values for both types of engines, we can compare them. The specific impulse value for the model rocket engine is 81.55 seconds, while the specific impulse value for the best chemical rocket engines is 407.75 seconds. The specific impulse of the best chemical rocket engines is much higher than that of the model rocket engine. This means that the best chemical rocket engines are much more efficient in utilizing their fuel/oxidizer per unit of ground weight, compared to the model rocket engines.