MEAM.Design - MEAM 247 - P1: Launch - Fundamental Rocket Relationships

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**Air Expansion**

We will begin with the rocket motor, filled with a certain volume of water, V_w(0), and air, V_a(0), at an initial pressure, P_a(0). Upon launch, we will assume adiabatic expansion of the air as water is quickly ejected through the nozzle. The pressure and volume of the air are thus related by:

where \gamma =1.4 is the adiabatic index for air.

**Water Nozzle Exit Velocity**

Recognizing that the water is essentially incompressible, we can use Bernoulli's equation to find the velocity of the water through the nozzle. Bernoulli's equation states that at any point in a streamline

where v is the velocity, p is the pressure, \rho is the density, g is gravity, and h is the height above a reference plane. Given that the motor is relatively short, we can ignore gravitational effects. We can therefore apply Bernoulli's equation to compare the nozzle exit to the air/water interface:

where v_e is the nozzle exit velocity, P_{atm} is the atmospheric pressure, and v_a is the velocity of the air/water boundary.. Now, solving for v_e:

**Thrust**

The linear momentum of a water particle of mass \Delta m expelled from the nozzle at velocity of v_e(t) can be expressed as

dividing by \Delta t and taking the limit as \Delta t goes to zero results in

which is equal to the sum of external forces acting on the water particle, and therefore equal to the thrust acting on the rocket

**Drag**

Courtesy of Dr. Kothmann, here are some notes on aerodynamic drag that you might find useful:

The Velocity Dependence of Aerodynamic Drag: A Primer for Mathematicians

Lyle Long and Howard Weiss

The American Mathematical Weekly, Vol. 106, No. 2. (Feb 1999)

pp 127-135