To determine the least velocity of the exhaust gas required for the rocket to just lift off, we can use the principles of Newton's second law and the rocket equation. The condition for lifting off is that the thrust generated by the rocket must be equal to or greater than the weight of the rocket including its propellant.
Given:
- Mass of the rocket (\( m_0 \)): 1000 kg
- Mass of the propellant gas (\( m_{propellant} \)): 3000 kg
- Total mass at lift-off (\( m_t \)): \( m_0 + m_{propellant} = 1000 , \text{kg} + 3000 , \text{kg} = 4000 , \text{kg} \)
- Rate of fuel consumption (\( \dot{m} \)): 60 kg/s
Steps to Calculate Least Velocity of Exhaust Gas (\( v_e \)):
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Calculate the weight of the rocket at lift-off: \[ W = m_t \cdot g \] where \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)).
\[ W = 4000 , \text{kg} \cdot 9.81 , \text{m/s}^2 = 39240 , \text{N} \]
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Calculate thrust required to lift off: To just lift off, the thrust (\( F \)) must equal the weight \( W \): \[ F = W = 39240 , \text{N} \]
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Calculate thrust generated by exhaust gas: The thrust produced by the rocket is given by: \[ F = \dot{m} \cdot v_e \] Rearranging for the velocity of exhaust gas: \[ v_e = \frac{F}{\dot{m}} \]
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Substituting the values: \[ v_e = \frac{39240 , \text{N}}{60 , \text{kg/s}} = 654 , \text{m/s} \]
Conclusion:
The least velocity of the exhaust gas (\( v_e \)) required for the rocket to just lift off is 654 m/s.