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The Rocket Thrust Equation is given by

where $\dot{m}$ is the mass flow rate, $v_e$ is the exhaust velocity, $p_e$ is the exhaust pressure, $p_0$ is the ambient pressure, and $A_e$ is the exhaust area. It is used to determine the thrust produced by a rocket engine.

The Specific Impulse Equation is given by

where $I_{sp}$ is the specific impulse, $F$ is the thrust, $\dot{m}$ is the mass flow rate, and $g_0$ is the standard gravity. It's a measure of the efficiency of rocket and jet engines.

The Burnout Velocity Equation is determined by combining the Tsiolkovsky rocket equation with the initial mass, propellant mass, and final mass:

where $v_b$ is the burnout velocity, $v_e$ is the exhaust velocity, $m_i$ is the initial mass, and $m_p$ is the propellant mass. It is used to calculate the velocity a rocket reaches after it has consumed all its propellant.

Delta-V Budget is estimated via the summation

where $\Delta v_{total}$ is the total change in velocity needed for a mission, and $\Delta v_i$ are the individual velocity changes for each segment of the trajectory. It is critical for mission planning and fuel requirements.

Similar to its use in aircraft, the Breguet Range Equation for Rockets can be expressed as

where $R$ is the range, $v_e$ is the exhaust velocity, $g_0$ is standard gravity, $L$ is the lift, $D$ is the drag, and $m_0$ and $m_f$ are the initial and final mass. It is useful for estimating the range a rocket can achieve given certain parameters.

The Ideal Rocket Equation is identical to the Tsiolkovsky Rocket Equation and is given by

where $I_{sp}$ is the specific impulse, $g_0$ is standard gravity. It's used to calculate the change in velocity in terms of specific impulse.

For continuous thrust trajectories, the spacecraft's velocity change is given by

where $v(t)$ is the velocity at time $t$, $v_0$ is the initial velocity, $F$ is the constant thrust, $m$ is the spacecraft mass, and $t_0$ is the initial time. This equation can be used for estimating flight trajectories under constant thrust acceleration.

The Tsiolkovsky rocket equation, given by

describes the maximum change in velocity a rocket can achieve without external forces. It relates the mass of propellant, exhaust velocity, and the initial and final mass of the rocket.

The Thrust-to-Weight Ratio is calculated by

where $TWR$ is the thrust-to-weight ratio, $F$ is the thrust, $m$ is the mass of the object, and $g$ is the acceleration due to gravity. It indicates whether a rocket can overcome gravitational forces to lift off.

The Mass Ratio Equation is given by

where $R$ is the mass ratio, $m_0$ is the initial mass, and $m_f$ is the final mass of the rocket. It's used to evaluate the proportion of mass that is propellant versus payload and structure.

The Spacecraft Maneuvering Equation, for a small maneuver, is given by

where $\Delta v$ is the change in velocity, $a$ is the acceleration provided by the spacecraft's engines, and $\Delta t$ is the duration of the burn. This is a simplified version used for quick calculations of velocity change due to short thrusts.

The Gravity Drag Equation is given by

where $\Delta v_{gd}$ is the change in velocity due to gravity drag, $g_0$ is standard gravity, and $t_{burn}$ is the burn time. It accounts for the velocity loss in a rocket due to gravity.

The Payload Fraction Equation is given by

where $\lambda$ is the payload fraction, $m_{payload}$ is the payload mass, and $m_0$ is the initial total mass. This metric helps in the design and efficiency analysis of the rocket.

The Exhaust Velocity Equation, derived from thermodynamic principles, is

where $v_e$ is the exhaust velocity, $k$ is the specific heat ratio, $T_c$ is the combustion temperature, and $M$ is the molar mass of the exhaust gases. It is used to determine the velocity of gas exiting a rocket nozzle.