Originally Posted by Kevbo
Solid fuel rockets often have thrust that varies as the surface area of the propellant changes. To avoid this requires that the propellant be divided into chunks that decrease in length as they increase in inside diameter. It is much simpler to make a single piece of propellant, but very difficult to get it to make constant thrust.
Although it is true that thrust on solid propellant rocket motors varies over the action time, the above explanation of how it is moderated isn't quite right. It is true that the thrust of a solid propellant rocket is essentially a function of two things; the propellant burn rate, and the exposed surface grain. The burn rate is a function of pressure with the burn rate coefficient (a, sometimes called the temperature coefficient as it is temperature-dependent) and the pressure exponent (n). It is generally desired (at least in large motors) to obtain high initial thrust with the thrust curve declining gently until it gets to the "knee" where the majority of propellant is burned and thrust drops off rapidly, called "tail-off". This is done by forming contours in the initial internal bore to control the exposed surface and pressure within the grain. (Note that while lower temperature means a lower burn rate and thus lower instantaneous thrust, it will also mean a longer action time, so total impulse is still essentially the same.) Depending on the size and shape of the grain, this may be done with radial grooves, star-shaped longitudinal features (fins), changes in the bore diameter, or combinations thereof, like finocyl (fins and cylinder change). However, in nearly all cases except small cartage-loaded tactical rockets and large segmented boosters, the propellant is cast in one continuous grain, not separate chunks, as any cracks or separation of grain that expose large amounts of surface will result in pressure spikes that can cause catastrophic feedback, as experienced on the Titan 34D-9 failure (due to a segmented grain that was not properly insulated and inhibited).
Even liquid propellant rocket engines, however, do not deliver constant thrust. While they are more controllable real-time during flight, they'll have an initial period of thrust ramp-up, followed by a period in which thrust will vary between certain parameters (due to combustion instability and shock losses) as well as expansion rate variation as ambient pressure decreases (assuming the rocket is ascending in atmosphere), and a tail-off period after pumps and valves are shut down while residual propellant is drawn from the propellant lines into the combustion chamber. Not understanding this can get you into a lot of trouble, as in the staging collision that occurred on the Falcon 1-2 flight. You may also throttle back the engines during particular regimes of flight, as the Shuttle does just prior to max-Q (maximum aerodynamic pressure) or to coast quasi-ballistically in order to optimize a trajectory profile.
is correct: you get total impulse by integrating thrust over time. For a simple thrust profile this is simple; you just draw the trapezoid that has the thrust profile outlined in the o.p., and then cut it into a rectangle and triangle, then solve by trivial geometry. The result should be in pound-force-seconds or newton-seconds (or poundal-seconds or dyne-seconds if you really want to be abstruse). Real world total impulse is generally calculated from accelerometer data and/or pressure traces.