Answer
The fraction of the initial mass that must be burned and ejected is $~~0.0022$
Work Step by Step
We can use Equation (9-88) to find the final mass $M_f$ in terms of the initial mass $M_i$:
$v_f-v_i = v_{rel}~ln(\frac{M_i}{M_f})$
$2.2~m/s = (1000~m/s)~ln(\frac{M_i}{M_f})$
$\frac{2.2~m/s}{1000~m/s} = ln(\frac{M_i}{M_f})$
$0.0022 = ln(\frac{M_i}{M_f})$
$e^{0.0022} = \frac{M_i}{M_f}$
$M_f = \frac{M_i}{e^{0.0022}}$
$M_f = 0.9978~M_i$
We can find the fraction of the initial mass that must be burned and ejected:
$M_i-M_f = M_i-0.9978~M_i = 0.0022~M_i$
The fraction of the initial mass that must be burned and ejected is $~~0.0022$
