By F. R. Gantmakher, L. M. Levin and H. L. Dryden (Auth.)
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2(Vl)], (63) [Fz(v)dv. (64) where L3(v)= o We could proceed indefinitely to obtain higher approximations for v. However for the majority of practical applications it is not necessary to go beyond the second approximation [equation (56)], and in the case of solid propellent field rockets we can very often be content simply with the Tsiolkovskii formula, for speeds u p to 350-400 m/sec (see Example 1 below). To determine the correction to the path-length we integrate equation (56): t % = h-~JL1(v1)dt-^f.
Formula (28) can be written in another form. TT. (29) 11. Variation in the Rocket Velocity during Powered Flight. Determination of the Arc-length of the Trajectory To ascertain the m a x i m u m velocity v& it was sufficient to know the ratio — = ^ a and the specific impulse j x . e. to determine v as a function of t, it is necessary to know X (t) and, consequently, the weight of charge burnt u p to the instant t. I n other words, the variation of v as a function of t is linked with the character of the combustion process.
Suppose t h a t t h e graph of a(t) is of t h e form shown in Fig. 10. We assume t h a t t h e launching velocity v is k n o w n : it can be determined from Fig. 10 (v0 = area of OA0B0C0). Let us imagine t h e uniformly accelerated motion produced backward in time from t h e i n s t a n t of launchingf (Fig. 10,— broken lines). We deduce t h e "effective'' values of t0 and s0: £0eff = — ,