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8.1 Raketa Asoslari — Masalalar

Jami: 25 ta | Yechim bilan: ✅

Asosiy Masalalar (1-10)​

Masala 1 ⭐⭐​

Raketa: m0=1000m_0 = 1000 kg, mf=300m_f = 300 kg, ve=3000v_e = 3000 m/s. Delta-v?

Yechim

Δv=veln⁡m0mf=3000ln⁡1000300=3000ln⁡(3.33)=3000×1.204=3612 m/s\Delta v = v_e \ln\frac{m_0}{m_f} = 3000 \ln\frac{1000}{300} = 3000 \ln(3.33) = 3000 \times 1.204 = 3612 \text{ m/s}

Masala 2 ⭐⭐​

Isp=300I_{sp} = 300 s. Exhaust tezligi?

Yechim

ve=Isp×g0=300×9.81=2943v_e = I_{sp} \times g_0 = 300 \times 9.81 = 2943 m/s ≈ 2940 m/s

Masala 3 ⭐⭐​

LEO uchun Δv=9.4\Delta v = 9.4 km/s kerak. ve=3000v_e = 3000 m/s. Mass ratio?

Yechim

MR=eΔv/ve=e9400/3000=e3.13=22.9MR = e^{\Delta v / v_e} = e^{9400/3000} = e^{3.13} = 22.9

Bu juda katta — shuning uchun ko'p bosqich kerak!

Masala 4 ⭐⭐​

Thrust = 1000 N, m˙=0.4\dot{m} = 0.4 kg/s. Exhaust tezligi?

Yechim

F=m˙⋅veF = \dot{m} \cdot v_e

ve=Fm˙=10000.4=2500v_e = \frac{F}{\dot{m}} = \frac{1000}{0.4} = 2500 m/s

Masala 5 ⭐⭐​

Raketa: 500 kg yonilg'i, burn time 50 s. Massa oqimi?

Yechim

m˙=mfueltburn=50050=10\dot{m} = \frac{m_{fuel}}{t_{burn}} = \frac{500}{50} = 10 kg/s

Masala 6 ⭐⭐​

Model raketa: C6-5 motor, total impulse 10 Ns, burn time 1.7 s. O'rtacha thrust?

Yechim

Favg=Itotaltburn=101.7=5.9F_{avg} = \frac{I_{total}}{t_{burn}} = \frac{10}{1.7} = 5.9 N

Masala 7 ⭐⭐​

Orbital tezlik 400 km balandlikda? (Yer radiusi 6371 km)

Yechim

r=6371+400=6771r = 6371 + 400 = 6771 km = 6.771×1066.771 \times 10^6 m

μ=GM=3.986×1014\mu = GM = 3.986 \times 10^{14} m³/s²

v=μr=3.986×10146.771×106=7672v = \sqrt{\frac{\mu}{r}} = \sqrt{\frac{3.986 \times 10^{14}}{6.771 \times 10^6}} = 7672 m/s ≈ 7.67 km/s

Masala 8 ⭐⭐​

Qochish tezligi Yerdan?

Yechim

vesc=2μR=2×3.986×10146.371×106=11186v_{esc} = \sqrt{\frac{2\mu}{R}} = \sqrt{\frac{2 \times 3.986 \times 10^{14}}{6.371 \times 10^6}} = 11186 m/s ≈ 11.2 km/s

Masala 9 ⭐⭐​

Gravity loss: burn time 150 s, vertical flight. Yo'qotish?

Yechim

Δvloss=g×t=10×150=1500\Delta v_{loss} = g \times t = 10 \times 150 = 1500 m/s = 1.5 km/s

Masala 10 ⭐⭐​

Model raketa: massa 200 g, motor thrust 8 N. Tezlanish (erkin)?

Yechim

Fnet=F−mg=8−0.2×10=6F_{net} = F - mg = 8 - 0.2 \times 10 = 6 N

a=Fnetm=60.2=30a = \frac{F_{net}}{m} = \frac{6}{0.2} = 30 m/s² = 3g

O'rtacha Masalalar (11-18)​

Masala 11 ⭐⭐⭐​

Ikki bosqichli raketa: har biri Δv=4\Delta v = 4 km/s. Umumiy Δv\Delta v?

Yechim

Δvtotal=Δv1+Δv2=4+4=8\Delta v_{total} = \Delta v_1 + \Delta v_2 = 4 + 4 = 8 km/s

(Staging foydaliroq chunki bo'sh tanklar tashlanadi)

Masala 12 ⭐⭐⭐​

1-bosqich: m0=100m_0 = 100 t, mf=30m_f = 30 t, ve=2800v_e = 2800 m/s. 2-bosqich: m0=25m_0 = 25 t, mf=5m_f = 5 t, ve=3200v_e = 3200 m/s. Umumiy Δv\Delta v?

Yechim

Δv1=2800ln⁡(100/30)=2800×1.204=3371\Delta v_1 = 2800 \ln(100/30) = 2800 \times 1.204 = 3371 m/s

Δv2=3200ln⁡(25/5)=3200×1.609=5149\Delta v_2 = 3200 \ln(25/5) = 3200 \times 1.609 = 5149 m/s

Δvtotal=3371+5149=8520\Delta v_{total} = 3371 + 5149 = 8520 m/s ≈ 8.5 km/s

Masala 13 ⭐⭐⭐​

Max-Q: v = 500 m/s, ρ = 0.5 kg/m³, CD=0.5C_D = 0.5, A = 10 m². Drag?

Yechim

q=12ρv2=0.5×0.5×5002=62500q = \frac{1}{2}\rho v^2 = 0.5 \times 0.5 \times 500^2 = 62500 Pa = 62.5 kPa

FD=q×CD×A=62500×0.5×10=312500F_D = q \times C_D \times A = 62500 \times 0.5 \times 10 = 312500 N = 312.5 kN

Masala 14 ⭐⭐⭐​

Raketa 1 km/s ga 30 s da yetishi kerak (vertikal). Kerakli thrust? (m = 1000 kg)

Yechim

Kerakli tezlanish: a=vt=100030=33.3a = \frac{v}{t} = \frac{1000}{30} = 33.3 m/s²

Gravitatsiyaga qarshi: atotal=a+g=33.3+10=43.3a_{total} = a + g = 33.3 + 10 = 43.3 m/s²

F=ma=1000×43.3=43300F = ma = 1000 \times 43.3 = 43300 N = 43.3 kN

Masala 15 ⭐⭐⭐​

Hohmann transfer: LEO (400 km) dan GEO (35786 km) ga. Δv\Delta v?

Yechim
import numpy as np

mu = 3.986e14  # m³/s²
R = 6.371e6    # m

r1 = R + 400e3
r2 = R + 35786e3

# LEO velocity
v1 = np.sqrt(mu/r1)

# Transfer orbit
a = (r1 + r2) / 2
v_trans_peri = np.sqrt(mu * (2/r1 - 1/a))
v_trans_apo = np.sqrt(mu * (2/r2 - 1/a))

# GEO velocity
v2 = np.sqrt(mu/r2)

# Delta-v
dv1 = v_trans_peri - v1
dv2 = v2 - v_trans_apo

print(f"Δv1 (LEO escape): {dv1:.0f} m/s")  # ~2440 m/s
print(f"Δv2 (GEO insert): {dv2:.0f} m/s")  # ~1470 m/s
print(f"Total: {dv1+dv2:.0f} m/s")         # ~3910 m/s

Masala 16 ⭐⭐⭐​

Payload fraction: Δv=9\Delta v = 9 km/s, ve=4v_e = 4 km/s, structure fraction = 10%.

Yechim

MR=e9/4=e2.25=9.49MR = e^{9/4} = e^{2.25} = 9.49

mf=m0/MRm_f = m_0 / MR

Structure = 10% of propellant stage

mstructure=0.1×(m0−mf)=0.1×m0(1−1/MR)m_{structure} = 0.1 \times (m_0 - m_f) = 0.1 \times m_0(1 - 1/MR)

mpayload=mf−mstructure=m0/MR−0.1m0(1−1/MR)m_{payload} = m_f - m_{structure} = m_0/MR - 0.1 m_0(1 - 1/MR)

λ=mpayloadm0=19.49−0.1(1−19.49)=0.105−0.089=0.016\lambda = \frac{m_{payload}}{m_0} = \frac{1}{9.49} - 0.1(1 - \frac{1}{9.49}) = 0.105 - 0.089 = 0.016

Payload fraction ≈ 1.6%

Masala 17 ⭐⭐⭐​

Thrust vectoring: 5° gimbal, thrust 100 kN. Lateral force?

Yechim

Flateral=Fsin⁡θ=100000×sin⁡5°=100000×0.087=8716F_{lateral} = F \sin\theta = 100000 \times \sin 5° = 100000 \times 0.087 = 8716 N ≈ 8.7 kN

Masala 18 ⭐⭐⭐​

Model raketa apogee: burnout velocity 50 m/s, drag coefficient 0.5. Taxminiy balandlik?

Yechim

Dragsiz: h=v22g=250020=125h = \frac{v^2}{2g} = \frac{2500}{20} = 125 m

Drag bilan (taxminan 30% yo'qotish):

hactual≈125×0.7=87.5h_{actual} \approx 125 \times 0.7 = 87.5 m ≈ ~90 m

Murakkab Masalalar (19-25)​

Masala 19 ⭐⭐⭐⭐​

Optimal staging: 2 bosqich, umumiy Δv=8\Delta v = 8 km/s, ve=3v_e = 3 km/s (har ikkisi). Har bir bosqich Δv\Delta v?

Yechim

Optimal: teng Δv\Delta v har bosqichda (structural ratio teng bo'lsa).

Δv1=Δv2=4\Delta v_1 = \Delta v_2 = 4 km/s

Mass ratio har biri: MR=e4/3=3.79MR = e^{4/3} = 3.79

Umumiy MR: 3.79×3.79=14.43.79 \times 3.79 = 14.4

(Bir bosqichli bo'lsa: MR=e8/3=14.4MR = e^{8/3} = 14.4 — bir xil!)

Lekin staging foydali chunki bo'sh struktura tashlanadi.

Masala 20 ⭐⭐⭐⭐​

Raketa simulyatsiya: vertikal uchish, drag bilan.

Yechim
import numpy as np
import matplotlib.pyplot as plt

def rocket_simulation():
    # Parameters
    m0 = 1000       # Initial mass (kg)
    m_prop = 700    # Propellant mass (kg)
    thrust = 15000  # N
    Isp = 280       # s
    burn_time = m_prop * 9.81 * Isp / thrust  # s
    
    Cd = 0.5
    A = 1           # m²
    
    # State
    m = m0
    v = 0
    h = 0
    
    dt = 0.1
    t = 0
    
    history = {'t': [], 'h': [], 'v': [], 'm': []}
    
    while h >= 0 or t < 1:
        # Atmosphere (exponential)
        rho = 1.225 * np.exp(-h / 8500)
        
        # Drag
        drag = 0.5 * rho * v**2 * Cd * A * np.sign(v)
        
        # Thrust
        if t < burn_time:
            F_thrust = thrust
            mdot = thrust / (Isp * 9.81)
            m -= mdot * dt
        else:
            F_thrust = 0
        
        # Acceleration
        a = (F_thrust - drag) / m - 9.81
        
        # Update
        v += a * dt
        h += v * dt
        t += dt
        
        history['t'].append(t)
        history['h'].append(h)
        history['v'].append(v)
        history['m'].append(m)
        
        if h < 0 and t > burn_time:
            break
    
    print(f"Burn time: {burn_time:.1f} s")
    print(f"Max altitude: {max(history['h']):.0f} m")
    print(f"Max velocity: {max(history['v']):.0f} m/s")
    
    return history

history = rocket_simulation()

Masala 21 ⭐⭐⭐⭐​

Orbital insertion: 200 km orbit, Δvtotal\Delta v_{total} = 9.5 km/s. Yonilg'i qancha kerak? (payload 1000 kg, ve=3.5v_e = 3.5 km/s, structure 8%)

Yechim
import numpy as np
from scipy.optimize import fsolve

def rocket_equation(m0, m_payload, ve, dv, struct_frac):
    # m0 = m_payload + m_structure + m_propellant
    # m_f = m_payload + m_structure
    # structure = struct_frac * m_propellant
    
    # Let x = m_propellant
    # m0 = m_payload + struct_frac * x + x = m_payload + x(1 + struct_frac)
    # m_f = m_payload + struct_frac * x
    
    # dv = ve * ln(m0/m_f)
    # Solve for x
    
    def equation(x):
        m0 = m_payload + x * (1 + struct_frac)
        m_f = m_payload + struct_frac * x
        return ve * np.log(m0 / m_f) - dv
    
    x_sol = fsolve(equation, 10000)[0]
    return x_sol

m_prop = rocket_equation(
    m0=None, 
    m_payload=1000, 
    ve=3500, 
    dv=9500, 
    struct_frac=0.08
)

m_struct = 0.08 * m_prop
m0 = 1000 + m_prop + m_struct

print(f"Propellant: {m_prop:.0f} kg")      # ~12700 kg
print(f"Structure: {m_struct:.0f} kg")     # ~1016 kg
print(f"Total mass: {m0:.0f} kg")          # ~14716 kg
print(f"Payload fraction: {1000/m0*100:.1f}%")

Masala 22 ⭐⭐⭐⭐​

Moon landing Δv\Delta v budget.

Yechim
SegmentΔv\Delta v (km/s)
LEO insertion9.4
Trans-lunar injection3.1
Lunar orbit insertion0.8
Descent & landing1.9
Total to Moon surface15.2
Ascent to lunar orbit1.9
Trans-Earth injection0.8
Return total2.7

Aerobraking Yerda — qo'shimcha Δv\Delta v kerak emas.

Masala 23 ⭐⭐⭐⭐​

Nozzle design: throat area, exit area, expansion ratio.

Yechim
import numpy as np

def nozzle_design(mdot, p_chamber, T_chamber, gamma=1.2, M_exit=3):
    """
    Simple nozzle sizing
    """
    R = 287  # J/(kg·K) for air-like gas
    
    # Throat conditions (M=1)
    T_throat = T_chamber * 2 / (gamma + 1)
    p_throat = p_chamber * (2 / (gamma + 1))**(gamma / (gamma - 1))
    rho_throat = p_throat / (R * T_throat)
    v_throat = np.sqrt(gamma * R * T_throat)
    
    A_throat = mdot / (rho_throat * v_throat)
    
    # Exit conditions
    T_exit = T_chamber / (1 + (gamma-1)/2 * M_exit**2)
    p_exit = p_chamber * (T_exit / T_chamber)**(gamma / (gamma-1))
    rho_exit = p_exit / (R * T_exit)
    v_exit = M_exit * np.sqrt(gamma * R * T_exit)
    
    A_exit = mdot / (rho_exit * v_exit)
    
    expansion_ratio = A_exit / A_throat
    
    return {
        'A_throat': A_throat,
        'A_exit': A_exit,
        'expansion_ratio': expansion_ratio,
        'v_exit': v_exit
    }

result = nozzle_design(
    mdot=100,           # kg/s
    p_chamber=7e6,      # Pa (70 bar)
    T_chamber=3500      # K
)

print(f"Throat area: {result['A_throat']*1e4:.1f} cm²")
print(f"Exit area: {result['A_exit']*1e4:.1f} cm²")
print(f"Expansion ratio: {result['expansion_ratio']:.1f}")
print(f"Exit velocity: {result['v_exit']:.0f} m/s")

Masala 24 ⭐⭐⭐⭐​

Stability margin: CP = 50 cm (nose'dan), CG = 35 cm, caliber = 5 cm.

Yechim

Stability margin = (CP - CG) / caliber

SM=50−355=3SM = \frac{50 - 35}{5} = 3 calibers

3 caliber — yaxshi barqarorlik!

(1-2 caliber minimal tavsiya)

Masala 25 ⭐⭐⭐⭐​

Recovery system: parachute sizing.

Yechim
import numpy as np

def parachute_size(mass, v_descent=5, Cd=1.5, rho=1.225):
    """
    Calculate parachute diameter for given descent rate
    """
    # Drag = Weight at terminal velocity
    # 0.5 * rho * v² * Cd * A = m * g
    
    A = (mass * 9.81) / (0.5 * rho * v_descent**2 * Cd)
    diameter = np.sqrt(4 * A / np.pi)
    
    return diameter, A

# Model rocket recovery
mass = 0.5  # kg
target_velocity = 5  # m/s

d, A = parachute_size(mass, target_velocity)
print(f"Parachute diameter: {d*100:.1f} cm")
print(f"Area: {A*1e4:.0f} cm²")

# Real rocket
mass = 1000
d, A = parachute_size(mass, v_descent=7)
print(f"\nCapsule chute diameter: {d:.1f} m")

✅ Tekshirish Ro'yxati​

  • 1-10: Tsiolkovsky va asosiy formulalar
  • 11-18: Staging va orbital mechanics
  • 19-25: Murakkab dizayn masalalari

Keyingi Qadam​

🔬 Amaliyot — Raketa simulyatsiyasi!