The rocket equation and delta-v

A cikin 1903 wani malamin makarantar da ya yi rauni a Kaluga, Konstantin Tsiolkovsky, ya rubuta ƙaddarorin da ke ci gaba da girman kowane roketi. Yana bayyana dalilin da ya sa wani Falcon 9 ne game da 89% propellant, me ya sa dauke shi ne kawai kaɗan cikin dari na nauyin a liftoff, kuma me ya sa rokets rasa wasu sassa na kansu a kan hanyar zuwa sama.

Duk wani roketi ne gasar tsakanin da sauri kana so da kuma nauyin da ka kamata a yi don su samu shi.

Look

Wani roketi yana tafiya ne da jefar da wani abu zuwa bãya. Ka jẽfa, kõ ka ƙãra wani abu daga gare shi, idan kã so ka kasance mafi gaggãwar tafiya.” Akwai wani kama: mai da ba ka shafe har yanzu dole ne kuma a kai da kuma gaggauta. Wancan ne yake saukar da alhẽri. logarithmic: don ƙara irin wannan adadin gudun, dole ne ka ninka mai, ba ka ƙara shi ba.

Delta-v (Δv, “canje a cikin gudu”) shine kudin shiga na jirgin sama. Duk wani jujjuyawar yana da farashinsa cikin kilomita/s:

• Fuskar ƙasa zuwa orbit mai ƙananan: kusan 9.4 km / s, gami da asarar.
• low orbit to geostationary orbit: about 3.9 km/s.
• low orbit to leave Earth for Mars: about 3.6 km/s.

Kudin jirgin sama shi ne yadda yawa Δv na injiniyoyinsa, mai da kuma nauyin zai iya bayarwa.

Understand

Δv = I_sp · g₀ · ln(m₀ / m_f)

• I_sp: Ƙaramar ƙarfi, ma’aunin ingancin injin cikin sakan (gudun fitarwa an raba shi da g₀).
• g₀ = 9.80665 m/s², ma’auni na jujjuyawar duniyar.
• m₀: nauyin a farkon rubu’in; m_f: nauyin a ƙarshe.

Misali da mataki na biyu na roket ɗin simulation (Isp 348 s): cikakke, yana nauyin kusan tan 112 ciki har da nauyin 15-ton; ba tare da propeller ba, kusan tan 19. Δv = 348 × 9.81 × ln(112 / 19) ≈ 6.0 km/s. Mataki na farko yana ƙara sauran.

A matakin daya da wani nau’i na 10 da Isp 350 s ba kawai 7.9 km / s, ba da yawa ga orbit lokacin da asarar aka lissafa, kuma wani tsari wanda ne kawai 10% na duka ne yanzu haske sosai. Wannan shi ne dalilin da ya sa kusan kowace roket mai tafiyar hawa biyu ke amfani da matakai biyu ko fiye.

Master

Staging multiplies mass ratios. Domin wani biyu-mataki jirgin sama,

Δv_total = I₁ g₀ ln(m₁₁/m_f₁) + I₂ g₀ ln(m₀₂/m_f₂)

A inda ma’aunin farawa na mataki na biyu ya ƙunsa duk abin da mataki na farko ya jefa. Optimizing da rarraba tsakanin matakai (ga daidai Isp, kusan daidai Δv a kowace mataki) shi ne wani classic matsala: the payload fraction ya fadi exponentially tare da bukata Δv, wanda shi ne dalilin da ya sa rage da Δv zuwa orbit ta wasu ɗaruruwan m / s (launching gabas, kusa da equator) muhimmanci sosai.

The equation kuma bayyanar da manhajar na sake amfani da farko matakai: su tashi baya amfani da mai da za a in ba haka ba ya kasance da damar yi, ciniki kaɗan bisa dari na nauyin don ba a jefa daga cikin mafi tsada sashi na roket.