A rocket of mass 500kg carrying,4000kg of fuel is to be launched vertically . the fuel is consumd at a steaty rate of 50kg/s calculate the least velocity of the exhaust gases if the launching pad immediately after firing (g=10m/s)
To calculate the least velocity of the exhaust gases immediately after firing, we need to use the principle of conservation of momentum.
The rocket initially consists of the rocket body and the fuel. The total initial mass of the rocket is the sum of the rocket body mass (500 kg) and the fuel mass (4000 kg). Therefore, the initial mass, M_initial, is 4500 kg.
As the fuel is consumed at a steady rate of 50 kg/s, after time t seconds, the remaining mass of the fuel will be 4000 kg - 50 kg/s * t.
According to the principle of conservation of momentum, the change in momentum of the rocket body and the fuel is equal to the change in momentum of the exhaust gases.
The momentum of an object is given by the product of its mass and velocity. So, the initial momentum of the rocket is the initial mass multiplied by the initial velocity, which is zero as it is initially at rest.
The final momentum of the rocket can be calculated by summing the momentum of the rocket body plus the momentum of the remaining fuel and the momentum of the exhaust gases.
The momentum of the rocket body after firing is its mass (500 kg) multiplied by its velocity (V_rb).
The momentum of the remaining fuel after firing is the remaining fuel mass multiplied by its velocity, which is zero as it is no longer contributing to the rocket's motion.
The momentum of the exhaust gases is the mass of the fuel consumed (50 kg/s * t) multiplied by the velocity of the exhaust gases (V_eg).
So the final momentum of the rocket is 500 kg * V_rb + (50 kg/s * t) * V_eg.
According to the principle of conservation of momentum, the initial momentum and the final momentum should be equal.
Therefore, 0 = 500 kg * 0 + (50 kg/s * t) * V_eg.
To find the least velocity of the exhaust gases, we need to find the value of V_eg. Since 50 kg/s * t is the mass of the fuel consumed, we can rewrite the equation as:
0 = (50 kg/s * t) * V_eg.
Since the mass of the fuel consumed, 50 kg/s * t, does not equal zero, we can divide both sides of the equation by (50 kg/s * t):
0 / (50 kg/s * t) = V_eg.
Simplifying further, we find:
0 = V_eg.
Therefore, the least velocity of the exhaust gases immediately after firing is zero, which means the exhaust gases are not ejecting from the rocket yet.