A rocket has an exhaust gas velocity of 2329 m/s and an expelled gas rate of 6761 kg/s. If it has a mass of 1.82 × 10^6 kg what is its initial acceleration ?
According to Newton's Second Law
acceleration = (Thrust-Weight)/Mass
The Thrust of the rocket is
T =(exhaust velocity)*(mass loss rate)
= 1.57*10^7 N
(That ranks it among the largest rockets)
Weight = 1.78*10^7 N
a = 0.21*10^7/(1.82*10^6) = 1.1 m/s^2
That is the initial acceleration rate. The acceleration rate increases as mass is lost as exhaust.
To find the initial acceleration of the rocket, we can use the principle of conservation of momentum.
The principle of conservation of momentum states that the change in momentum of an object is equal to the net force acting on it multiplied by the time taken for the force to act. In this case, the net force acting on the rocket is equal to the force exerted by the expelled gas.
The momentum of the rocket is given by the product of its mass and velocity. The momentum of the expelled gas is given by the product of its mass rate of expulsion and velocity.
In this case, the mass rate of expulsion is given as 6761 kg/s and the velocity of the expelled gas is given as 2329 m/s.
The momentum of the rocket can be expressed as:
Momentum = Mass × Velocity
= (1.82 × 10^6 kg) × 0 m/s
= 0 kg*m/s
The momentum of the expelled gas can be expressed as:
Momentum = Mass Rate of Expulsion × Velocity of the Expelled Gas
= (6761 kg/s) × (2329 m/s)
= 1.57369 × 10^7 kg*m/s
According to the principle of conservation of momentum, the change in momentum of the system (rocket + expelled gas) is equal to zero. Therefore, the momentum of the rocket and the expelled gas before and after the expulsion should cancel each other out.
Change in Momentum = Momentum of Rocket + Momentum of Expelled Gas
0 = 0 kg*m/s + 1.57369 × 10^7 kg*m/s
0 = 1.57369 × 10^7 kg*m/s
From the equation above, we can conclude that the momentum of the rocket and the expelled gas before and after the expulsion is the same.
Now, we can calculate the initial acceleration of the rocket using the equation:
Acceleration = Force / Mass
The force exerted by the expelled gas is equal to the momentum of the expelled gas divided by the time taken for the expulsion.
Force = Momentum of Expelled Gas / Time Taken for Expulsion
= (1.57369 × 10^7 kg*m/s) / (Time Taken for Expulsion)
We don't have the specific time taken for the expulsion, so we are unable to directly calculate the force. However, we can rearrange the equation to solve for acceleration without the need for the time taken for expulsion.
Acceleration = Force / Mass
= (Momentum of Expelled Gas / Time Taken for Expulsion) / Mass
= (Momentum of Expelled Gas / Mass) / Time Taken for Expulsion
Substituting the given values for mass and the momentum of the expelled gas, we have:
Acceleration = (1.57369 × 10^7 kg*m/s) / (1.82 × 10^6 kg) / Time Taken for Expulsion
= 8.6366 m/s² / Time Taken for Expulsion
Therefore, the initial acceleration of the rocket is 8.6366 m/s² divided by the time taken for the expulsion. However, since we don't have the specific time taken for the expulsion, we cannot determine the exact value of the initial acceleration.