Problem 7 - Aircraft Propulsion

Question

A rocket is vertically launched and operates for \(60 \mathrm{~s}\) and has a mass ratio of \(0.05\). The (mean) rocketspecific impulse is \(375 \mathrm{~s}\). Assuming the average gravitational acceleration over the burn period is \(9.70 \mathrm{~m} / \mathrm{s}^{2}\), calculate the terminal velocity of the rocket with and without gravitational effects. Neglect the effect of aerodynamic drag in both cases.

Step-by-Step Solution

Answer

The terminal velocity of the rocket with gravitational effects is approximately 6651.1 m/s and without gravitational effects is approximately 8233.11 m/s.

Step 1: Parameters Identification

Identify the parameters given in the problem. The duration for which the rocket operates is 60 s, the mass ratio is 0.05, the rocket's specific impulse is 375 s, and the average gravitational acceleration is 9.70 m/s². The mass ratio implies that 95% of the rocket's initial mass is the propellant.

Step 2: Formula derivation with Gravitational Effects

The terminal velocity of the rocket considering gravitational effects can be derived using the rocket equation: \(V_f = I_{sp} * g_0 * ln(1/MR) - g_0 * t_b\), where \(V_f\) is the final velocity, \(I_{sp}\) is the specific impulse, \(g_0\) is the standard gravitational acceleration, MR is the mass ratio, and \(t_b\) is the burn time.

Step 3: Calculation with Gravitational Effects

Plug in the given values and calculate the final velocity, \(V_{f1} = 375 * 9.81 * ln(1/0.05) - 9.70 * 60 = 6651.1 \ m/s\)

Step 4: Formula derivation without Gravitational Effects

The terminal velocity of the rocket disregarding gravitational effects can be calculated using the rocket equation without the term involving the burn time: \(V_f = I_{sp} * g_0 * ln(1/MR)\).

Step 5: Calculation without Gravitational Effects

Plug in the given values and calculate the final velocity, \(V_{f2} = 375 * 9.81 * ln(1/0.05) = 8233.11 \ m/s\).

Key Concepts

Specific Impulse

In rocket propulsion, specific impulse ( I_{sp} ) is a crucial concept that measures how efficiently a rocket uses its propellant. It is defined as the impulse per unit weight of propellant. This metric helps us understand how much thrust can be obtained from a given amount of propellant and is usually expressed in seconds. Think of specific impulse as the 'fuel economy' of the rocket engine—higher values indicate more efficient propellant usage. In our context, a specific impulse of 375 seconds is used, implying that each unit weight of propellant produces thrust for 375 seconds in standard gravity conditions. Given this efficiency measure, one can appreciate why specific impulse is central to calculating a rocket's performance during its mission. When specific impulse increases, less propellant is needed for the same velocity, enabling longer or more resource-intensive missions.

Mass Ratio

The mass ratio ( MR ) of a rocket defines the proportion of the rocket's initial launch mass that is propellant. It's a critical factor in determining how much of the rocket's fuel is used up during the flight. In the context of this exercise, the mass ratio is given as 0.05. This means that only 5% of the total weight remains once all the propellant has been burned, indicating that 95% of the initial mass was propellant. This concept matters because the mass ratio directly influences the velocity the rocket can achieve. A greater mass ratio implies a greater percentage of mass is fuel, which can lead to higher potential velocities. The mass ratio affects the calculations of both gravitational and non-gravitational conditions, as more propellant translates to greater thrust and longer burn durations.

Gravitational Effects

Gravitational effects are essential to consider when calculating a rocket's velocity, specifically in vertical launches. Gravity acts against the thrust of the rocket, slowing it down. In our exercise, the gravitational acceleration over the burn period is slightly less than the standard Earth gravity, at 9.70 m/s². Gravitational effects reduce the final velocity of the rocket as they impose a continuous downward force during propellant burn time. When we account for gravitational effects in the rocket equation, it modifies how quickly the rocket can accelerate to its final speed. Ignoring these effects, you get a theoretical maximum velocity. Including gravity provides a more realistic speed as the constant opposing force is considered.

Rocket Equation

The rocket equation is at the heart of rocket motion calculations and governs how the velocity changes based on various factors. This equation, also known as the Tsiolkovsky rocket equation, is expressed as:\[ V_f = I_{sp} \times g_0 \times \ln(1/MR) - g_0 \times t_b \]where \(V_f\) is the terminal velocity, \(I_{sp}\) the specific impulse, \(g_0\) the standard gravitational acceleration, \(MR\) the mass ratio, and \(t_b\) the burn time. This formula allows us to calculate the final velocity of a rocket under various conditions, whether we're accounting for gravity or ignoring it. The logarithm term \(\ln(1/MR)\) reflects the exponential effect that the mass ratio has on the velocity. The subtraction of \(g_0 \times t_b\) adjusts the velocity calculation for the gravitational force experienced during the propulsion phase. Understanding and applying the rocket equation are critical in determining a vehicle's performance based on its propellant efficiency and operating conditions.