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3/65 The rocket moves in a vertical plane and is being propelled by a thrust T of 32 kN. It is also subjected to an atmospheric resistance R of 9.6 kN. If the rocket has a velocity of 3 km/s and if the gravitational acceleration is 6 m/s² at the altitude of the rocket, calculate the radius of curvature p of its path for the position described and the time-rate-of-change of the magnitude v of the velocity of the rocket. The mass of the rocket at the instant considered is 2000 kg.

3/65 The rocket moves in a vertical plane and is being propelled by a thrust T of 32 kN. It is also subjected to an atmospheric resistance R of 9.6 kN. If the rocket has a velocity of 3 km/s and if the gravitational acceleration is 6 m/s² at the altitude of the rocket, calculate the radius of curvature p of its path for the position described and the time-rate-of-change of the magnitude v of the velocity of the rocket. The mass of the rocket at the instant considered is 2000 kg.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Please solve for:

-Magnitude of acceleration of the rocket in m/s^2
-Direction of acceleration of the rocket relative to the ground in degrees

Box these answers, the unknowns that are asked for in the problem statement aren't necessary.

v = 6 m/s^2
p = 3000 km

 

**Problem 3/65: Rocket Motion in a Vertical Plane** A rocket moves in a vertical plane and is propelled by a thrust \(T\) of 32 kN. It is also subjected to an atmospheric resistance \(R\) of 9.6 kN. The rocket has a velocity of 3 km/s. Given that the gravitational acceleration is 6 m/s\(^2\) at the altitude of the rocket, calculate the following: 1. The radius of curvature \(\rho\) of its path for the described position. 2. The time-rate-of-change of the magnitude \(v\) of the velocity of the rocket. At the instant considered, the mass of the rocket is 2000 kg. ### Given Data: - Thrust, \(T = 32\) kN - Atmospheric resistance, \(R = 9.6\) kN - Velocity, \(v = 3\) km/s = 3000 m/s - Gravitational acceleration, \(g = 6\) m/s\(^2\) - Mass of the rocket, \(m = 2000\) kg ### Calculation: 1. **Radius of Curvature (\(\rho\)):** The radius of curvature \(\rho\) is dependent on the centripetal force acting on the rocket. The necessary centripetal force can be derived from the gravitational and thrust forces minus the drag force. 2. **Time-Rate-of-Change of Velocity (\(\dot{v}\)):** The time-rate-of-change of the magnitude of velocity of the rocket can be determined using Newton's second law, applying the net force along the direction of the velocity vector. The details of these computations would typically involve resolving the forces in the vertical direction and applying principles of dynamics to solve for the respective quantities. Piecing together the forces (thrust, drag, and gravity) and solving for the unknowns using the appropriate dynamical equations will yield the radius of curvature and the rate-of-change of velocity. This problem illustrates the application of dynamics in an aerospace context, combining forces due to propulsion, resistance, and gravity to determine the motion characteristics of a rocket.
Transcribed Image Text:**Problem 3/65: Rocket Motion in a Vertical Plane** A rocket moves in a vertical plane and is propelled by a thrust \(T\) of 32 kN. It is also subjected to an atmospheric resistance \(R\) of 9.6 kN. The rocket has a velocity of 3 km/s. Given that the gravitational acceleration is 6 m/s\(^2\) at the altitude of the rocket, calculate the following: 1. The radius of curvature \(\rho\) of its path for the described position. 2. The time-rate-of-change of the magnitude \(v\) of the velocity of the rocket. At the instant considered, the mass of the rocket is 2000 kg. ### Given Data: - Thrust, \(T = 32\) kN - Atmospheric resistance, \(R = 9.6\) kN - Velocity, \(v = 3\) km/s = 3000 m/s - Gravitational acceleration, \(g = 6\) m/s\(^2\) - Mass of the rocket, \(m = 2000\) kg ### Calculation: 1. **Radius of Curvature (\(\rho\)):** The radius of curvature \(\rho\) is dependent on the centripetal force acting on the rocket. The necessary centripetal force can be derived from the gravitational and thrust forces minus the drag force. 2. **Time-Rate-of-Change of Velocity (\(\dot{v}\)):** The time-rate-of-change of the magnitude of velocity of the rocket can be determined using Newton's second law, applying the net force along the direction of the velocity vector. The details of these computations would typically involve resolving the forces in the vertical direction and applying principles of dynamics to solve for the respective quantities. Piecing together the forces (thrust, drag, and gravity) and solving for the unknowns using the appropriate dynamical equations will yield the radius of curvature and the rate-of-change of velocity. This problem illustrates the application of dynamics in an aerospace context, combining forces due to propulsion, resistance, and gravity to determine the motion characteristics of a rocket.
**Problem 3/65** This image depicts a rocket that is inclined at an angle of 30° from the vertical axis. The rocket's orientation is illustrated with the body slightly tilted to the left. Two vectors, denoted as \( R \) (at the top of the rocket) and \( T \) (at the bottom of the rocket near the thrusters), are shown extending outward. Both vectors are depicted with red arrows, indicating the directions they represent. - **Angle Measurement**: The rocket's orientation is measured at 30° from the vertical axis. - **Vector \( R \)**: Located at the top of the rocket, pointing outward from the body. - **Vector \( T \)**: Located at the bottom of the rocket near the thrusters, pointing outward in the opposite direction. The purpose of this problem likely involves analyzing the forces and moments acting on the rocket. Understanding these vectors' directions in the diagram will be critical for solving the related physics or engineering problem.
Transcribed Image Text:**Problem 3/65** This image depicts a rocket that is inclined at an angle of 30° from the vertical axis. The rocket's orientation is illustrated with the body slightly tilted to the left. Two vectors, denoted as \( R \) (at the top of the rocket) and \( T \) (at the bottom of the rocket near the thrusters), are shown extending outward. Both vectors are depicted with red arrows, indicating the directions they represent. - **Angle Measurement**: The rocket's orientation is measured at 30° from the vertical axis. - **Vector \( R \)**: Located at the top of the rocket, pointing outward from the body. - **Vector \( T \)**: Located at the bottom of the rocket near the thrusters, pointing outward in the opposite direction. The purpose of this problem likely involves analyzing the forces and moments acting on the rocket. Understanding these vectors' directions in the diagram will be critical for solving the related physics or engineering problem.
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