The terms v and m are the velocity of the rocket and its mass at any point in time thereafter (respectively). The variables are defined as follows: v i is the initial rocket velocity and m i is the initial rocket mass. This is a very useful equation coming out of the analysis, shown above. Integrate the above equation using Calculus. Thus,Īgain, the term v+ v e is the velocity of the exhaust gases relative to the rocket, which is approximately constant. The mass of the ejected rocket exhaust equals the negative of the mass change of the rocket. The maximum acceleration is therefore just before all the propellant burns off. The term dm e/ dt is the burn rate of rocket propellant.Īs the rocket loses mass due to the burning of propellant, its acceleration increases (for a given thrust T). This is approximately constant in rockets. The term v+ v e is the velocity of the exhaust gases relative to the rocket. Therefore, the thrust T acting on the rocket is equal to The left side of this equation must represent the thrust acting on the rocket, since a = dv/dt is the acceleration of the rocket, and Σ F = ma (Newton's second law). Since the rocket is moving in a vacuum, with no gravity, and no air resistance (drag), then Σ F y = 0 since no external forces are acting on the system. In the limit as dt→0 we may neglect the "second-order" term dm edv. Where Σ F y is the sum of the external forces in the vertical direction acting on all the particles in the system (consisting of rocket and exhaust).Įxpand the above expression. We can express this mathematically using Calculus and the principle of impulse and momentum: The sign convention in the vertical direction is as follows: "up" is positive and "down" is negative.īetween (1) and (2), the change in linear momentum in the vertical direction of all the particles in the system, is due to the sum of the external forces in the vertical direction acting on all the particles in the system. Note that all velocities are measured with respect to ground (an inertial reference frame). V e is the velocity of the exhaust exiting the rocket, at stage (2) This remains constant between (1) and (2)ĭm e is the mass of rocket propellant that has exited the rocket (in the form of exhaust), between (1) and (2)ĭv is the change in velocity of the rocket, between (1) and (2)
J e is the linear momentum of the rocket exhaust (that has already exited the rocket), at stage (1). V is the velocity of the rocket, at stage (1) M e is the total mass of the rocket exhaust (that has already exited the rocket), at stage (1) M is the mass of the rocket (including propellant), at stage (1) The system (consisting of rocket and exhaust) is shown as inside the dashed line. The two stages, (1) and (2), show the "state" of the system at time t and time t+ dt, where dt is a very small (infinitesimal) time step. To properly analyze the physics, consider the figure below which shows a schematic of a rocket moving in the vertical direction. For simplicity, we will assume the rocket is moving in a vacuum, with no gravity, and no air resistance (drag). In this analysis of the rocket physics we will use Calculus to set up the governing equations. To find the equations of motion, apply the principle of impulse and momentum to the "system", consisting of rocket and exhaust. The equations of motion of a rocket will be derived next. This is the essential principle behind the physics of rockets, and how rockets work. This backwards acceleration of the exhaust exerts a "push" force on the rocket in the opposite direction, causing the rocket to accelerate forward. As the rocket propellant ignites, it experiences a very large acceleration and exits the back of the rocket (as exhaust) at a very high velocity. A rocket has variable mass because its mass decreases over time, as a result of its fuel (propellant) burning off.Ī rocket obtains thrust by the principle of action and reaction (Newton's third law). Rocket physics, in the most basic sense, involves the application of Newton's Laws to a system with variable mass. Rocket Physics Picture of Saturn V Launch for Apollo 15 Mission.
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