###### Orbital Mechanics for Engineering Students

146 SOLVED PROBLEMS

Question: 11.7

## Find the optimal mass for a three-stage launch vehicle that is required to lift a 5000 kg payload to a speed of 10 km/s. For each stage, we are given that  Stage 1  Isp1 = 400 s (c1 = 3.924 km/s) ε1 = 0.10 Stage 2  Isp2 = 350 s (c2 = 3.434km/s) ε2 = 0.15 Stage 3  Isp3 = 300 s (c3 = 2.943 km/s) ε3 = ...

Substituting this data into Equation 11.86, we get...
Question: 11.6

## (a) Find the extrema of the function z = – x² – y². (b) Find the extrema of the same function under the constraint y = 2x + 3. ...

(a) To find the extrema we must use Equations 11.6...
Question: 11.5

## Repeat Example 11.4 for the restricted three-stage launch vehicle. ...

Equation 11.56 gives the burnout velocity for thre...
Question: 11.4

## The following data are given mPL = 10,000 kg πPL = 0.05 ε = 0.15 ISP = 350s go = 0.00981 km/s² (a) Calculate the payload velocity vbo at burnout, the empty mass of the launch vehicle and the propellant mass for (a) a single stage and (b) a restricted, two-stage vehicle. ...

(a) From Equation 11.43 we find v_{b o}=I_{...
Question: 11.3

## The rocket in Example 11.2 has a diameter of 5 m. It is to be launched on a gravity turn trajectory. Pitchover begins at an altitude of 130 m with an initial flight path angle γo of 89.85° . What are the altitude h and speed v of the rocket at burnout (tbo = 260 s)? What are the velocity losses ...

The $\text { MATLAB }^{\circledR}$ pr...
Question: 11.2

## The data for a single stage rocket are as follows: Launch mass: mo = 68,000 kg Mass ratio: n = 15 Specific impulse: Isp = 390 s Thrust: T = 933.91 kN It is launched into a vertical trajectory, like a sounding rocket. Neglecting drag and assuming that the gravitational acceleration is constant at ...

(a) From Example 11.1(b) the burnout time t...
Question: 11.1

## A sounding rocket of initial mass mo and mass mf after all propellant is consumed is launched vertically ( γ = 90°). The propellant mass flow rate me is constant. (a) Neglecting drag and the variation of gravity with altitude, calculate the maximum height h attained by the rocket. (b) For what ...

The vehicle mass as a function of time, up to burn...
Question: 10.15

## The uniform, monolithic 10,000 kg slab, having the dimensions shown in Figure 10.31 , is in a circular LEO. Determine the orientation of the satellite in its orbit for gravity gradient stabilization, and compute the periods of the pitch and yaw/roll oscillations in terms of the orbital period T . ...

Let us first determine whether we can stabilize th...
Question: 10.14

## A satellite is in torque-free motion. A control moment gyro, spinning at the constant rate ωs, is gimbaled about the spacecraft y and z axes, with φ = 0 and φ = 90° (cf. Figure 10.26 ). The spacecraft angular velocity is ω = ωzk. If the spin axis of the gyro, initially along the x direction, is ...

Substituting \omega_{x}=\omega_{y}=\dot{\om...