how fast does a rocket need to go to break earth's gravitationl pull
The final rocket velocity required to maintain a circular orbit around the Earth may be computed from the following:
Vc = sqrt(µ/r)
where Vc is the circular orbital velocity in feet per second, µ (pronounced meuw as opposed to meow) is the gravitational constant of the earth, ~1.40766x10^16 ft.^3/sec.^2, and r is the distance from the center of the earth to the altitude in question in feet. Using 3963 miles for the radius of the earth, the orbital velocity required for a 250 miles high circular orbit would be Vc = 1.40766x10^16/[(3963+250)x5280] = 1.40766x10^16/22,244,640 = 25,155 fps. (17,147 mph.) Since velocity is inversely proportional to r, the higher you go, the smaller the required orbital velocity.
The fundamental flight stages of a rocket may be defined as Pre-Launch, liftoff, vertical rise, pitchover, ascent, staging (1st stage burnout-2nd stage ignition), ascent, staging (2nd stage burnout-3rd stage ignition), ascent, final burnout. With ignition of the 1st stage rocket motors, the vehicle rises vertically from the launch pad for some predetermined number of seconds, usually 2 or 3. This time period is referred to as the vertical rise time, VRT. At the end of the VRT, the rocket executes a pitchover maneuver where the flight path takes on a slight angle to the vertical, again, usually a couple of degrees, and usually referred to as the "kick angle." From this point on, most launch rockets now fly what is termed a gravity turn, or zero angle of attack, trajectory, to reduce the aerodynamic forces on the vehicle. Throughout this phase, the rocket thrust is directed along the velocity vector and the force of gravity continuously pitches the rocket over as it ascends. The flight path angle at burnout of the 1st stage is a direct function of the initial kick angle at liftoff. At 1st and 2nd stage burnout, the rocket is considered to be sufficiently out of the atmosphere such that these stages can now be flown on a pitch program (the thrust vector and the velocity vector no longer coincide) without regard to aerodynamic effects. Finally, after the rocket has reached its desired altitude, velocity, and flight path angle, the last stage is shutdown, and the vehicle is in orbit. For a 100 mile high circular orbit, the vehicle velocity must be 25,616 ft./sec. and the flight path must be perpendicular to the line through the Earth's center and the vehicle. Some additional information regarding what a rocket must deal with during launch follows.
A rocket lifting off from the surface of the earth must constantly fight three primary elements that offer resistance to its flight; gravity, drag, and atmospheric pressure. Rockets rise by means of the thrust provided by the rocket engines. By definition, the thrust from the rocket engines must be greater than the weight of the vehicle being lifted or it wouldn't move off the launch pad. Unfortunately, at liftoff, and for a considerable time period after liftoff, the rocket vehicle is not able to derive full benefit from the thrust of the rocket engines as gravity is continuously trying to negate the acceleration of the vehicle provided by the thrust. As the rocket pitches over during the ascent, the effect of gravity is continuously reduced by the cosine of the flight path angle to the local vertical. Eventually, as the rocket approaches its target altitude, its flight path, or velocity direction, is approaching a perpendicular to a line radiating from the center of the earth, tangent to the circular or elliptical orbit into which the rocket is being injected. At this time, the affect of gravity on the rocket's acceleration is essentially zero.
Another factor that a rocket encounters during launch is the drag force on the vehicle produced by the dynamic pressure of the atmosphere on the nose of the vehicle. This drag is directly proportional to the air density, the frontal area of the rocket, and the square of the velocity. The major contributor to the total drag force is the nose which steadily pierces the atmosphere at increasing velocity as the rocket climbs through, and out of, the atmosphere. Even though the nose is usually cone shaped to reduce drag, it is this surface that is fighting its way through the air. A secondary, though sometimes predominant, contributor to the total drag, is the friction drag produced along the entire length of the vehicle. The total drag force may be calculated from the equation D = Cd(d)AV^2/2 where Cd (see sub d) is the total integrated drag coefficient of the vehicle, derived from wind tunnel tests, d is the mass density of the atmosphere, A is the frontal area of the rocket, and V is the velocity. When the rocket is first lifting off, the velocity is low but the air density is highest and as the velocity increases with increasing altitude, the air density decreases. However, the drag increases with the square of the velocity so has an overriding effect on the magnitude of the drag. Ultimately, the air density drops to the point where air drag is no longer a concern.
Lastly, the thrust of a rocket, though theoretically constant, is actually reduced by approximately 15% at liftoff, due to the presence of sea level atmospheric pressure. As the rocket rises through the atmosphere, atmospheric pressure reduces, and the thrust from the motors increases by the difference in pressure times the exit area of the rocket motor nozzles. The rocket motors only reach their vacuum design thrust levels when they are out of the atmoshhere.
As stated earlier, another factor to be accounted for in determining the propulsive energy required for a rocket is the fact that a rocket launched from the surface of the earth actually starts out with, and retains, the velocity of the earth's surface at the launch site all the way to orbit. To overcome these gravity/drag resistances and atmospheric thrust losses, we must add additional increments of propulsive energy to our rocket to make up for these losses such that the net velocity gain from our rocket is sufficient to reach our final desired velocity.
The actual flight path of a rocket is a function of the thrust to weight ratio of the rocket and the vehicle's ballistic coefficient, CdA/W, where Cd and A are as defined earlier and W is the weight of the rocket. The primary goal in deriving a trajectory profile is to minimize the total of the gravity and drag losses. This optimization procedure is always a struggle between the gravity and drag losses. The drag losses can be minimized by maintaining a vertical flight path and getting out of the atmosphere as fast as possible. However, the longer the rocket remains vertical, the higher the gravity losses. On the other hand, the sooner the rocket
pitches over and steadily increases its flight path angle to the vertical, the lower the gravity losses. But, this keeps the rocket in the atmosphere longer and thereby increases the drag losses. Since the gravity losses make up the majority of the total, preference is usually given to pitching the rocket over as soon as possible into what is called a gravity turn trajectory, thus allowing the drag losses to become higher but reducing the gravity losses to compensate.
Most of today's launch rockets consist of a minimum of two stages. The first stage contributes most of the energy losses due to its being in the atmosphere the longest and flying the closest to the vertical the longest. The only loss the second stage incurs is primarily gravity losses and even this is a minimum as the second stage is approaching the final flight path angle, normal to the perpendicular through the center of the earth. The total of the launch losses typically amounts to approximately 20-25% of the the net velocity gain needed from the rocket's propulsion system. For example, a rocket requires a final burnout velocity of 25,155 fps. to be inserted into a 250 mile high circular orbit. When launching from Cape Canaveral, Florida, at ~28.5 degrees north latitude, the contribution of the earth's rotational velocity is ~1340 fps so the rocket must only provide a net velocity gain of 25,155 - 1340 = 23,815 fps. The average losses for a typical, medium thrust to weight, rocket to this altitude would be in the neighborhood of ~5,400 fps with ~75-80% of this coming from gravity losses, 10-15% coming from drag losses, and the remainder from thrust losses.
In summary, gravity and the atmosphere, will forever be the enemy of the rocket designer. It is common to try and design a rocket with a high thrust to weight ratio so that its higher acceleration will enable it to rise through the atmosphere as fast as possible and approach its final velocity and flight path as quickly as possible. Manned rockets however, are typically limited in the level of thrust to weight ratio they can fly at due to the "g" limitations on the flight crew.