Question

A spaceship's orbital maneuver requires a speed increase of 1.35 ✕ 10³ m/s. If its engine has an exhaust speed of 2.25 ✕ 10³ m/s, determine the required ratio M_i/M_f

 of its initial mass to its final mass. (The difference M_i − M_f

 equals the mass of the ejected fuel.)

____.

Answer 1

According to Raw q conservation q momentum we have:

$(M+Δm)V = M(v+Δv)+Δm(v-v_e) \\ mΔv=v_eΔm \\ Δm=-ΔM \\ MΔv =-v_eΔM \\ v_f-v_i =v_eln(\frac{M_i}{M_f}) \\ \frac{v_f-v_i}{v_e} = ln(\frac{M_i}{M_f}) \\ \frac{1.35 \times 10^3}{2.25 \times 10^3} = ln(\frac{M_i}{M_f}) \\ \frac{M_i}{M_f} = e^{\frac{1.35}{2.25}} = e^{0.6}=1.822$