Physics

Physics · mechanics

Problem

A spacecraft is orbiting in a very low circular orbit at a velocity v0v_{0} over the equator of a perfectly spherical moon with uniform density. Relative to a stationary frame, the spacecraft completes a revolution of the moon every 90 minutes, while the moon revolves in the same direction once every 24 hours. The pilot of the spacecraft would like to land on the moon using the following process:So the final answer is 1. Start by firing the engine directly against the direction of motion.So the final answer is 2. Orient the engine over time such that the vertical velocity of the craft remains 0 , while the horizontal speed continues to decrease. 3. Once the velocity of the craft relative to the ground is also 0 , turn off the engine. Assume that the engine of the craft can be oriented instantly in any direction, and the craft has a TWR (thrust-to-weight ratio, where weight refers to the weight at the moon's surface) of 2, which remains constant throughout the burn. If the craft starts at v0=500 m/sv_{0}=500 \mathrm{~m} / \mathrm{s}, compute the delta-v expended to land, minus the initial velocity, i.e. Δv−v0\Delta v-v_{0}. Given constants: - Proton mass, mp=1.67⋅10−27 kgm_{p}=1.67 \cdot 10^{-27} \mathrm{~kg} - Neutron mass, mn=1.67⋅10−27 kgm_{n}=1.67 \cdot 10^{-27} \mathrm{~kg} - Electron mass, me=9.11⋅10−31 kgm_{e}=9.11 \cdot 10^{-31} \mathrm{~kg} - Avogadro's constant, N0=6.02⋅1023 mol−1N_{0}=6.02 \cdot 10^{23} \mathrm{~mol}^{-1} - Universal gas constant, R=8.31 J/(mol⋅K)R=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) - Boltzmann's constant, kB=1.38⋅10−23 J/Kk_{B}=1.38 \cdot 10^{-23} \mathrm{~J} / \mathrm{K} - Electron charge magnitude, e=1.60⋅10−19Ce=1.60 \cdot 10^{-19} \mathrm{C} - 1 electron volt, 1eV=1.60⋅10−19 J1 \mathrm{eV}=1.60 \cdot 10^{-19} \mathrm{~J} - Speed of light, c=3.00⋅108 m/sc=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s} - Universal Gravitational constant, G=6.67⋅10−11( N⋅m2)/kg2G=6.67 \cdot 10^{-11}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{kg}^{2} - Solar Mass M⊙=1.988⋅1030 kgM_{\odot}=1.988 \cdot 10^{30} \mathrm{~kg} - Acceleration due to gravity, g=9.8 m/s2g=9.8 \mathrm{~m} / \mathrm{s}^{2} - 1 unified atomic mass unit, 1u=1.66⋅10−27 kg=931MeV/c21 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}=931 \mathrm{MeV} / \mathrm{c}^{2} - Planck's constant, h=6.63⋅10−34 J⋅s=4.41⋅10−15eV⋅sh=6.63 \cdot 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.41 \cdot 10^{-15} \mathrm{eV} \cdot \mathrm{s} - Permittivity of free space, ϵ0=8.85⋅10−12C2/(N⋅m2)\epsilon_{0}=8.85 \cdot 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right) - Coulomb's law constant, k=14πϵ0=8.99⋅109( N⋅m2)/C2k=\frac{1}{4 \pi \epsilon_{0}}=8.99 \cdot 10^{9}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{C}^{2} - Permeability of free space, μ0=4π⋅10−7 T⋅m/A\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} - Magnetic constant, μ04π=1⋅10−7( T⋅m)/A\frac{\mu_{0}}{4 \pi}=1 \cdot 10^{-7}(\mathrm{~T} \cdot \mathrm{m}) / \mathrm{A} - 1 atmospheric pressure, 1 atm=1.01⋅105 N/m2=1.01⋅105 Pa1 \mathrm{~atm}=1.01 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}=1.01 \cdot 10^{5} \mathrm{~Pa} - Wien's displacement constant, b=2.9b=2.9. 10−3 m⋅K10^{-3} \mathrm{~m} \cdot \mathrm{K} - Stefan-Boltzmann constant, σ=5.67⋅10−8 W/m2/K4\sigma=5.67 \cdot 10^{-8} \mathrm{~W} / \mathrm{m}^{2} / \mathrm{K}^{4}

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