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Kinematic equations are
                                    x -. V                            (2.139)
                                                                h : 0                                               (2.140)
For range and endurance problems, we need an additional equation that describes
the variation of weighL This is obtained by considering fuel consumption. For a
jet airplane, the specific fuel consumption is the amount of fueJ consumed per unit
thrust per unit time so that the variation of weight is given by
W - -cT                           (2.141)
where c is the specific fuel consumption. For the specific fuel consumption ofjet
engines, we will use the units of N/Nh. Here, we assume that for a given altitude,
specific fuel consumption is a constant. In other words, variation of specific fuel
consumption with velocity is ignored. A negative sign is chosen in Eq. (2.141)
because weight decreases as fuel is consumed.
   Equation (2.139) can be written as
dx dW
dW dt = V                    (2.142)
dx    V
dW = -cT
         V
     ::= -                            (2.143)
   cD
The quantity dxld W is often called the instantaneous range and is equal to the hor-
izontal distance traversed per unit load of fuel or the specific range.It is analogous
to the gas mileage of an automobile.
    Multiply and divide the right-hand side ofEq. (2.143) by W and use the relation
L  - W to obtain
dx
dW =-(~) (,VW)
     = -(g ) ,VW                       (2.144)
108              PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Let R = xf  - xi denote the horizontal distance covered during the cruise. Then,
                           R = -[:' (g )(1  VdWW                  (2.145)
where Wo is the initi,al weight and Wi is the final weight. Let
                                          Wf .= Wo - Wi                              (2.146)
denote the weight of fuel consumed during the cruise.
      To integrate6Eq. (2.145), we have to specify the variation oflift and drag coeffi-
cients and the velocity during the cruise. We assume that the angle of. attack is held
constant at some value throughout the cruise so that the lift and drag coefficients are
constants. To determine this value of angle of attack and the corresponding flight
velocity, we consider two special cases as discussed in the following subsections.
   Range at constant altitude.  To hold the altitude at a constant value, the
velocit)t must varjr continuously during the cruise to compensate for the variation
in weight (because of fuel consumption) according to the relation,
V:
(2.147)
where W is the instantaneous weight of the airplane. Then, substitution in Eq.
　
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