A rocket of mass 1000 kg containing a propellant gas of 3000kg is to be launched vertically. If the fuel is consumed at a rate of 60kgs-1, calculate the least velocity of the exhaust gasses if the rocket and content will just lift off the launching pad immediately after firing
F = d/dt(mv) = m g = 4000 *9.81
v * 60kg/s = 4000 * 9.81
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666.67
To calculate the least velocity of the exhaust gases required for the rocket to lift off the launching pad, we need to apply Newton's second law of motion.
According to Newton's second law, the force required to lift the rocket off the ground is equal to the total mass of the system (rocket + propellant) multiplied by the acceleration:
Force = (m_rocket + m_propellant) * acceleration
In this case, the force required is the weight of the system, since we want to overcome the force of gravity:
Force = (m_rocket + m_propellant) * g
where g is the acceleration due to gravity (approximately 9.8 m/s²).
The force generated by the exhaust gases is equal to the rate of change of momentum (mass flow rate multiplied by the velocity of the exhaust gases):
Force_gas = rate_of_change_of_momentum = m_dot * v_gas
where m_dot is the rate at which the fuel is consumed (60 kg/s), and v_gas is the velocity of the exhaust gases.
To lift off the launching pad, the force generated by the exhaust gases must be equal to the weight of the system:
(m_rocket + m_propellant) * g = m_dot * v_gas
Now we can solve for v_gas:
v_gas = ((m_rocket + m_propellant) * g) / m_dot
Substituting the given values:
v_gas = ((1000 kg + 3000 kg) * 9.8 m/s²) / 60 kg/s
Calculating:
v_gas = (4000 kg * 9.8 m/s²) / 60 kg/s
v_gas ≈ 653.33 m/s
Therefore, the least velocity of the exhaust gases required for the rocket to lift off the launching pad immediately after firing is approximately 653.33 m/s.