or, integrating between an initial time to, and a final time t AU=U-U0=-c(In m-In mo)=-cIn Alternatively, this expression can be cast as the well known rocket equation m no e which gives the mass of the rocket at a time t, as a function of the mass initial mass mo, Au, and c. The mass of the propellant propellant, Is given by From the above equations, we see that for a given Aw and mo, increasing c increases m(payload) and decreases propellant. Unfortunately, we can only choose c as high as the current technology will allow Note that if ri is constant, we can write m(t)=mo +rit, and substitute into equation 6 to obtain an expression for v as a function of t, U=0-cln(1+ Recall that according to the convention used, i is negative as the mass decreases with time 2)Ft=-mg A constant gravitational field acting in the opposite direction to the velocity vector can be easily corporated. In this case, equation 5 become dt ng Thich can be integrated to give U=vo-cIn--gt= vo -cIn - no 3)Ft f--mg-D The effect of the drag force, D, is harder to quantify. It turns out that for many important applications drag effects are very small. The drag force is characterized in terms of a drag coefficient, CD. Thus, where A is the cross-sectional area of the rocket. The air density changes with altitude z, and may be approximated byor, integrating between an initial time t0, and a final time t, ∆v = v − v0 = −c(ln m − ln m0) = −c ln m m0 . (6) Alternatively, this expression can be cast as the well known rocket equation, m = m0 e −∆v/c , (7) which gives the mass of the rocket at a time t, as a function of the mass initial mass m0, ∆v, and c. The mass of the propellant, mpropellant, is given by, mpropellant = m0 − m = m0(1 − e −∆v/c). From the above equations, we see that for a given ∆v and m0, increasing c increases m (payload) and decreases mpropellant. Unfortunately, we can only choose c as high as the current technology will allow. Note that if ˙m is constant, we can write m(t) = m0 + ˙mt, and substitute into equation 6 to obtain an expression for v as a function of t, v = v0 − c ln(1 + m˙ m0 t) . Recall that according to the convention used, ˙m is negative as the mass decreases with time. 2) Ft = −mg A constant gravitational field acting in the opposite direction to the velocity vector can be easily incorporated. In this case, equation 5 becomes, m dv dt = −mg − c dm dt , which can be integrated to give v = v0 − c ln m m0 − gt = v0 − c ln m m0 − g m − m0 m˙ . 3) Ft = −mg − D The effect of the drag force, D, is harder to quantify. It turns out that for many important applications drag effects are very small. The drag force is characterized in terms of a drag coefficient, CD. Thus, D = 1 2 ρv2ACD , where A is the cross-sectional area of the rocket. The air density changes with altitude z, and may be approximated by ρ = ρ0e −z/H , 3