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)
Well, I must admit, this question has really "taken off!" So, let's calculate the least velocity of the exhaust gases, shall we?
The total mass of the rocket and the fuel is 500 kg + 4000 kg = 4500 kg. Now, as the fuel is consumed at a steady rate of 50 kg/s, we need to consider the time it takes to consume all the fuel.
Since the rocket has 4000 kg of fuel and it's consumed at a rate of 50 kg/s, it will take 4000 kg / 50 kg/s = 80 seconds to consume all the fuel.
Now, we can calculate the net force acting on the rocket. The force of gravity, Fg, can be found using the formula Fg = mg, where m is the mass of the rocket and g is the acceleration due to gravity (10 m/s²). Thus, Fg = 4500 kg * 10 m/s² = 45000 N.
The thrust force, Ft, exerted by the rocket is given by the equation Ft = (dm / dt) * Ve, where dm/dt is the rate of change of mass (fuel consumption rate) and Ve is the exhaust velocity.
Since the mass is consumed at a steady rate of 50 kg/s, dm/dt = 50 kg/s. Now, to calculate the least velocity of the exhaust gases, we assume the rocket is initially at rest. Therefore, the net force acting on the rocket is equal to the thrust force: Ft = Fg.
So, 45000 N = (50 kg/s) * Ve. Rearranging the equation, we find Ve = 45000 N / 50 kg/s = 900 m/s.
Therefore, the least velocity of the exhaust gases must be 900 m/s.
That's one "fired up" rocket, don't you think? Happy launching!
To calculate the least velocity of the exhaust gases, we can use the principle of conservation of momentum.
The total mass of the rocket and fuel is given as 500kg (rocket) + 4000kg (fuel) = 4500kg.
At any instant during the rocket's flight, the total momentum (p) is given by the product of total mass (m) and velocity (v).
Initially, before the rocket is launched, the momentum is zero as the rocket is at rest. Therefore, the total momentum after firing must also be zero for the rocket to achieve vertical motion.
When the fuel is consumed at a steady rate of 50kg/s, the mass of the rocket and fuel decreases as time passes. Let's assume that t seconds have elapsed since the fuel started to burn.
The mass of the rocket at time t is 500kg, and the mass of the consumed fuel is 50kg/s * t.
The mass of the remaining fuel at time t is 4000kg - (50kg/s * t).
Now, we can rewrite the conservation of momentum equation as follows:
0 = (500kg + [4000kg - (50kg/s * t)]) * V + ([50kg/s * t] * Ve)
Where:
V = velocity of the rocket
Ve = velocity of the exhaust gases
We need to calculate the least velocity of the exhaust gases, which means we want to find the minimum value for Ve. This occurs when the velocity of the rocket (V) is at its maximum.
Let's assume the time taken for all the fuel to be consumed is t_total seconds.
At this time, the remaining fuel mass will be 4000kg - (50kg/s * t_total) = 0kg.
Substituting these values into the equation, we get:
0 = (500kg + 0kg) * V + (0kg * Ve)
0 = 500V
To satisfy the equation, V must be zero, meaning the velocity of the rocket will be zero at the time when all the fuel is consumed.
Therefore, the least velocity of the exhaust gases immediately after firing is zero.
To calculate the least velocity of the exhaust gases immediately after firing, we can use the principle of conservation of momentum. The momentum of the rocket and its fuel before firing is equal to the momentum of the rocket and exhaust gases after firing.
The initial momentum of the system (before firing) is given by the sum of the momentum of the rocket and the momentum of the fuel:
Initial momentum = (mass of rocket + mass of fuel) * velocity
The final momentum of the system (after firing) is given by the momentum of the rocket and the momentum of the exhaust gases:
Final momentum = mass of rocket * velocity
Since there is no external force acting on the system in the vertical direction, the total momentum of the system is conserved.
Therefore, we can equate the initial momentum to the final momentum:
Initial momentum = Final momentum
(mass of rocket + mass of fuel) * velocity = mass of rocket * velocity
Simplifying the equation:
mass of rocket * velocity + mass of fuel * velocity = mass of rocket * velocity
mass of fuel * velocity = 0
Since the mass of the fuel is being consumed at a steady rate, we can determine the time it takes to consume all the fuel:
Time taken to consume all fuel = mass of fuel / rate of fuel consumption
Time taken = 4000 kg / 50 kg/s
= 80 seconds
During these 80 seconds, the velocity of the exhaust gases must be zero.
After the fuel is exhausted, the velocity of the rocket is determined by:
mass of rocket * final velocity = mass of rocket * initial velocity + mass of exhaust gases * exhaust velocity
Since the final velocity of the rocket is the least velocity of the exhaust gases immediately after firing, we can set the mass of the exhaust gases and the final velocity of the rocket to zero:
mass of rocket * final velocity = 0
Therefore, the least velocity of the exhaust gases immediately after firing is zero.