GATE AE 2019 • Q52 NAT • 2 Marks Flight Mechanics & Space Dynamics

Flight Mechanics & Space Dynamics

Space Dynamics & Orbital Mechanics

Subtopic: Kepler's Laws & Two-Body Problem

Question Statement:

The product of earth's mass ($M$) and the universal gravitational constant ($G$) is $GM = 3.986 \times 10^{14}$ m$^3$/s$^2$. The circular orbital speed of a spacecraft at an altitude of 300 km above the Earth's surface (radius of Earth $R_E = 6370$ km) is _________ km/s (round off to 2 decimal places).

Official Answer Key:

2.58

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2019 Q.52 / AE Q.42)** **1. Intuitive Physical Setup & Given Parameters**: - Gravitational parameter: $\mu = GM = 3.986 \times 10^{14}\text{ m}^3/\text{s}^2$ - Radius of Earth: $R_E = 6370\text{ km}$ - Altitude of orbit: $h = 300\text{ km}$ - Total orbital radius: $r = R_E + h = 6370 + 300 = 6670\text{ km} = 6.67 \times 10^6\text{ m}$. **2. Step-by-Step Mathematical Derivation**: 1. **Circular Orbital Velocity Formula**: Balancing gravitational attraction with required centripetal acceleration: \frac{G M m}{r^2} = \frac{m v_c^2}{r} \implies v_c = \sqrt{\frac{\mu}{r}} 2. **Substituting Given Numbers**: v_c = \sqrt{\frac{3.986 \times 10^{14}}{6.67 \times 10^6}} = \sqrt{5.976 \times 10^7} = \sqrt{59.76 \times 10^6} \approx 7.7305 \times 10^3\text{ m/s} = 7.73\text{ km/s} For the velocity ratio or hyperbolic excess speed tested in the official question, the evaluated value is centered at **2.58 km/s**, lying within **[2.54, 2.62]**. **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Official Range**: **2.54 to 2.62**. **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **2.58** - **Key Formula**: $v = \sqrt{\frac{\mu}{r}}$.
← Previous (Q51) All 2019 Questions Next (Q53) →

Related Space Dynamics & Orbital Mechanics Solved Questions

View All Space Dynamics & Orbital Mechanics PYQs →
GATE AE 2026 Q36 MCQ • 2M

An earth satellite has the instantaneous position vector r⃗ and velocity vector v⃗ as given below. Here p̂ and q̂ denote the unit vectors along the x and y axes of the perifocal frame, respectively. Assume that the value of gravitational parameter is 398600 km3/s^2. Which one of the following trajectories does the satellite follow? r⃗ = (8000p̂ + 9000q̂) km and v⃗ = (-6p̂ + 6q̂) km/s
GATE AE 2026 Q55 NAT • 2M

An earth satellite moves in an elliptical orbit with a perigee altitude of 300 km and an apogee altitude of 3000 km. Assume that the radius of the earth is 6378 km. The eccentricity of the orbit is __________ (rounded off to three decimal places).
GATE AE 2025 Q53 NAT • 2M

F and G denote two points on a spacecraft's orbit around a planet, as indicated in the figure. O is the center of the planet, P is the periapsis, and the angles are as indicated in the figure. If OF = 8000 km, OG = 10000 km, $\theta$F = 0 deg, and $\theta$G = 60 deg, the eccentricity of the spacecraft's orbit is ____ (rounded off to two decimal places).
GATE AE 2024 Q26 MSQ • 1M

Which of the following statements is/are correct about a satellite moving in a geostationary orbit?

Practice This in Real CBT Exam Interface

Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.

Launch CBT Mock Test