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
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 and also clearly state the magnitude of acceleration of the rocket in m/s^2 and direction of the rocket relative to the ground in degrees.
![### Problem Statement:
**Rocket Motion Analysis**
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. If the rocket has a velocity of 3 km/s and if the gravitational acceleration is 6 \( \text{m/s}^2 \) at the altitude of the rocket, calculate:
1. The radius of curvature \( \rho \) of its path for the position described.
2. 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.
### Definitions and Assumptions:
- The thrust \( T \) is the force propelling the rocket forwards.
- Atmospheric resistance \( R \) acts in the opposite direction to the motion.
- Gravitational acceleration affects the rocket by pulling it downwards.
- The mass of the rocket affects its acceleration in response to the forces acting on it.
### Given Data:
- Thrust, \( T = 32 \) kN
- Atmospheric resistance, \( R = 9.6 \) kN
- Velocity, \( v = 3 \) km/s
- Gravitational acceleration, \( g = 6 \text{ m/s}^2 \)
- Mass of the rocket, \( m = 2000 \) kg
### Calculations:
To find the radius of curvature \( \rho \) of the rocket's path, the formula used is:
\[ \rho = \frac{v^2}{a_c} \]
where \( a_c \) is the centripetal acceleration.
To find the time-rate-of-change of the magnitude of velocity \( v \), Newton's second law is applied:
\[ F_{net} = ma \]
**Note:** As the problem does not include graphical elements or diagrams, no further details on graphs or diagrams are provided.
### Analysis:
1. **Compute the net force acting on the rocket:**
\[ F_{net} = T - R - mg \]
2. **Determine the actual acceleration of the rocket:**
\[ a = \frac{F_{net}}{m} \]
3. **Calculate the radius of curvature \( \rho \):**
\[ \rho = \frac{v^2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda0a2617-650f-47df-ae27-c9852e282155%2Fd948635b-70e2-43e2-90e5-9345eb32bce2%2F654vxd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
**Rocket Motion Analysis**
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. If the rocket has a velocity of 3 km/s and if the gravitational acceleration is 6 \( \text{m/s}^2 \) at the altitude of the rocket, calculate:
1. The radius of curvature \( \rho \) of its path for the position described.
2. 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.
### Definitions and Assumptions:
- The thrust \( T \) is the force propelling the rocket forwards.
- Atmospheric resistance \( R \) acts in the opposite direction to the motion.
- Gravitational acceleration affects the rocket by pulling it downwards.
- The mass of the rocket affects its acceleration in response to the forces acting on it.
### Given Data:
- Thrust, \( T = 32 \) kN
- Atmospheric resistance, \( R = 9.6 \) kN
- Velocity, \( v = 3 \) km/s
- Gravitational acceleration, \( g = 6 \text{ m/s}^2 \)
- Mass of the rocket, \( m = 2000 \) kg
### Calculations:
To find the radius of curvature \( \rho \) of the rocket's path, the formula used is:
\[ \rho = \frac{v^2}{a_c} \]
where \( a_c \) is the centripetal acceleration.
To find the time-rate-of-change of the magnitude of velocity \( v \), Newton's second law is applied:
\[ F_{net} = ma \]
**Note:** As the problem does not include graphical elements or diagrams, no further details on graphs or diagrams are provided.
### Analysis:
1. **Compute the net force acting on the rocket:**
\[ F_{net} = T - R - mg \]
2. **Determine the actual acceleration of the rocket:**
\[ a = \frac{F_{net}}{m} \]
3. **Calculate the radius of curvature \( \rho \):**
\[ \rho = \frac{v^2
Transcribed Image Text:**Problem 3/65: Rocket Orientation**
This image depicts a rocket at an angle relative to the vertical axis. The rocket is inclined at an angle of 30° from the vertical direction.
Two forces are acting on the rocket:
- \( T \): This force is acting towards the tail end of the rocket.
- \( R \): This force is acting towards the nose of the rocket.
The force vectors \( T \) and \( R \) are represented by red arrows, indicating their directions and points of application. This scenario is presented as Problem 3/65, potentially asking for the analysis of forces, moments, and possibly the resulting motion or stability of the rocket at this given inclination.
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