2.3 Flight Dynamics
To accelerate the rocket, a force must be applied to it. Dynamics is the mechanics of forces, which cause the motion that kinematics describes. A basic law of motion, discovered by Isaac Newton, is that force, F, applied to a mass, m, causes it to accelerate:
F = m× a
When multiple external forces are applied to a mass, the resultant force is the vector sum of the individual force vectors, applied at the center of mass. The rocket engine applies force to the rocket, and it reacts by accelerating at the rate of F/m.
Another basic quantity of motion, momentum, is mass times velocity:
momentum = m× v
When two hard objects in motion bounce off each other, the sum of their momenta before and after is the same; momentum is conserved. For rockets starting at rest, the momentum of the rocket equals the momentum of the exhaust jet. The rocket engine exhaust has a much higher speed than the rocket (the exhaust speed, ve, but the mass of the rocket is much larger than the mass of the burnt propellant leaving the engine nozzle. Conservation of momentum in this case can be expressed by the equation:
In engine design, a rocket of a given mass goes faster if its exhaust jet has higher speed. Consequently, it is desirable to maximize exhaust speed. The propellant fraction of total rocket mass becomes the exhaust mass, me. By making this fraction as large as possible, me is maximized, thereby increasing vr. This fraction is called the propellant fraction and is the reciprocal of the mass ratio,
where ms is the mass of the rocket structure, that of the rocket minus the propellant mass, mp. In other words, mass ratio is the full initial launch mass, m0, over the empty mass. The higher the mass ratio, the greater the fraction of rocket mass (or weight) that is propellant and the lighter the empty rocket is.
Force is related to momentum. It is the rate of change of momentum, or
The D of a product can be derived (see "D(m× v)" box) to produce the more general force equation:
The dot over a quantity indicates rate of change of the quantity, or d(quantity)/dt.
The difference of a product can be found by the following algebraic derivation. A more rigorous derivation requires differential calculus.
First, note that
This can be written as
The propulsive force, or thrust, of a rocket is based on the thrust equation:
The first term is the reactive thrust of the engine. The second term is the pressure thrust. A higher nozzle exit pressure than that of the atmosphere exerts a force over the nozzle exit area, Ae. The resulting force due to pressure is F = P× A. The contribution of this term is usually less than a few percent of the total thrust. It is the push of the engine against the atmosphere.
The dominant contribution to thrust is from the reactive thrust, where is the total propellant mass flow rate and ve is the exit (or exhaust) speed of the jet leaving the nozzle. Our goal in engine design will be to maximize ve. (Here, ve is the component of the flow velocity from the nozzle exit in line with the longitudinal axis of the rocket.)
A common performance parameter of propulsion systems is the specific impulse, Isp;
where g0 is the gravitational acceleration on the earth’s surface, or 9.81 m/s2 (@ 32.2 ft/s2). Specific impulse has units of seconds; the higher the value, the better the performance. Typical LOX-kerosene engines have Isp values of about 200 s while the high-performance Space Shuttle hydrogen-oxygen engine achieves about 450 s. The more "energetic" the propellant, the higher the specific impulse, and consequently the higher the exhaust speed and thrust.
The exhaust speed, ve, is related to the specific impulse, Isp, which is a figure of merit of the propellant. If we neglect pressure thrust, Fp, then
and ve @ Isp× g0. But specific impulse includes the effect of pressure thrust. For an exact equation, the effective exhaust speed, c, is defined as:
In other words, c is ve plus whatever additional speed the exhaust would gain if pressure thrust were considered as additional reaction thrust. If total thrust were related to exhaust speed, the speed would be c, though the actual exhaust has speed ve.