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Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a mass
loss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity
the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!?
loss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity
the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!?
Most Upvoted Answer
Rockets are propelled by the momentum reaction of the exhaust gases ex...
Rocket Motion and Equation Derivation
The motion of a rocket in a uniform gravitational field can be modeled by considering the forces acting on it. The change in momentum from the expelled gases produces a reaction force according to Newton's third law.
- The equation of motion is given by:
m dv/dt = −v' (dm/dt) − mg
- m: mass of the rocket
- dv/dt: acceleration of the rocket
- v': velocity of the escaping gases relative to the rocket
- dm/dt: rate of mass loss
- g: acceleration due to gravity
Integration to Find Velocity
Assuming a constant rate of mass loss, we define:
- dm/dt = -k, where k is a positive constant.
Rearranging gives:
- dv = -\frac{v'}{m} k dt - g dt
Integrating this equation from initial mass m0 to final mass m and from time 0 to t leads to:
- v(m) = v' ln(m0/m) - gt
Escape Velocity Requirement
To achieve escape velocity, the rocket must reach a speed of approximately 11.2 km/s. Given:
- Initial mass loss rate k = \frac{m0}{60}
- v' = 2.1 km/s
Setting the condition for escape velocity:
- v' ln(m0/m) - gt = 11.2 km/s
Assuming the empty rocket mass is negligible, the ratio of the weight of the fuel (Wf) to the weight of the empty rocket (We) can be derived.
- The necessary ratio is approximately:
Wf/We ≈ 300
This indicates that for a rocket to achieve escape velocity, the fuel must weigh nearly 300 times more than the empty rocket itself, illustrating the significant energy required to overcome gravitational forces.
The motion of a rocket in a uniform gravitational field can be modeled by considering the forces acting on it. The change in momentum from the expelled gases produces a reaction force according to Newton's third law.
- The equation of motion is given by:
m dv/dt = −v' (dm/dt) − mg
- m: mass of the rocket
- dv/dt: acceleration of the rocket
- v': velocity of the escaping gases relative to the rocket
- dm/dt: rate of mass loss
- g: acceleration due to gravity
Integration to Find Velocity
Assuming a constant rate of mass loss, we define:
- dm/dt = -k, where k is a positive constant.
Rearranging gives:
- dv = -\frac{v'}{m} k dt - g dt
Integrating this equation from initial mass m0 to final mass m and from time 0 to t leads to:
- v(m) = v' ln(m0/m) - gt
Escape Velocity Requirement
To achieve escape velocity, the rocket must reach a speed of approximately 11.2 km/s. Given:
- Initial mass loss rate k = \frac{m0}{60}
- v' = 2.1 km/s
Setting the condition for escape velocity:
- v' ln(m0/m) - gt = 11.2 km/s
Assuming the empty rocket mass is negligible, the ratio of the weight of the fuel (Wf) to the weight of the empty rocket (We) can be derived.
- The necessary ratio is approximately:
Wf/We ≈ 300
This indicates that for a rocket to achieve escape velocity, the fuel must weigh nearly 300 times more than the empty rocket itself, illustrating the significant energy required to overcome gravitational forces.
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Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!?
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Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!?.
Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!?.
Solutions for Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!? in English & in Hindi are available as part of our courses for Physics.
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Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!?, a detailed solution for Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!? has been provided alongside types of Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is m dv/dt = −v' dm/dt − mg,where m is the mass of the rocket and v is the velocity of the escaping gases relative tothe rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket starting initially from rest, with v equal to 2.1 km/s and a massloss per second equal to 1/60th of the initial mass, that inorder to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!? theory, EduRev gives you an
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