An aeroplane of mass 3500kg of which the aviation fuel is 6000kg,if the fuel is consumed at 180kg/s.what is the minimum velocity of the exhaust gases to enable the plane take off
Then use conservation of momentum
Well, to determine the minimum velocity of the exhaust gases for a plane to take off, we need to consider that the force generated by the exhaust gases should equal the weight of the plane.
Now, let's do some calculations. Since we know the mass of the plane (3500 kg) and the acceleration due to gravity (9.8 m/s^2), we can calculate the weight of the plane, which is 3500 kg * 9.8 m/s^2 = 34,300 N.
Since all the fuel is consumed, the total mass of the plane and fuel is 3500 kg + 6000 kg = 9500 kg. The mass flow rate of the fuel is given as 180 kg/s.
Now, to find the minimum velocity of the exhaust gases, we can use the principle of conservation of momentum. The force generated by the exhaust gases can be calculated as the mass flow rate of the fuel times the velocity of the exhaust gases:
Force = mass flow rate * velocity
Since the force generated should equal the weight of the plane (34,300 N), we can set up the equation:
34,300 N = 180 kg/s * velocity
Solving for velocity, we find:
velocity = 34,300 N / 180 kg/s
velocity ≈ 190.56 m/s
So, the minimum velocity of the exhaust gases would need to be approximately 190.56 m/s for the plane to take off. Although, it might be best to confirm this with an actual pilot.
To determine the minimum velocity of the exhaust gases required for the plane to take off, we can use the principle of conservation of momentum.
1. First, we need to find the change in momentum of the plane when all the fuel is consumed. The change in momentum can be calculated using the formula:
Δp = m * Δv
where Δp is the change in momentum, m is the mass, and Δv is the change in velocity.
2. Since the aviation fuel is being consumed, the mass of the plane decreases over time. The remaining mass of the plane after the fuel is consumed is given by:
m_plane = m_initial - m_fuel
where m_plane is the mass of the plane, m_initial is the initial mass of the plane, and m_fuel is the mass of the fuel.
3. The change in momentum can be written as:
Δp = (m_initial - m_fuel) * Δv
4. The change in velocity can be calculated using the equation:
Δv = v_final - v_initial
where v_final is the final velocity of the plane after the fuel is consumed, and v_initial is the initial velocity of the plane.
5. To take off, the final velocity of the plane needs to be greater than or equal to zero. So, if we assume that the initial velocity is zero, we can write:
Δv = v_final
6. Now, let's substitute the values into the equation:
Δp = (m_initial - m_fuel) * Δv
Δv = v_final
Δp = (m_initial - m_fuel) * Δv
7. We can rearrange the equation to solve for Δv:
Δv = Δp / (m_initial - m_fuel)
8. The change in momentum is equal to the momentum of the exhaust gases. Momentum is given by the equation:
p = m * v
where p is the momentum, m is the mass, and v is the velocity.
9. Since we want the minimum velocity of the exhaust gases, we assume that the mass of the exhaust gases is zero after they leave the plane. Therefore, the change in momentum is equal to the momentum of the fuel:
Δp = m_fuel * v_exhaust
where v_exhaust is the velocity of the exhaust gases.
10. We can substitute this value into the equation and solve for the minimum velocity of the exhaust gases:
Δv = Δp / (m_initial - m_fuel)
Δv = (m_fuel * v_exhaust) / (m_initial - m_fuel)
To find the minimum velocity, we need the values for the initial mass (m_initial) and the mass of the fuel (m_fuel). Please provide the values for these variables.
To find the minimum velocity of the exhaust gases required for the plane to take off, we can use the principle of conservation of momentum. When the fuel is consumed and expelled as exhaust gases, there will be an equal and opposite force acting on the plane, providing the necessary thrust to overcome gravity and allow the plane to take off.
First, let's calculate the mass of the plane without fuel:
mass of the plane = total mass - mass of the fuel
mass of the plane = 3500 kg - 6000 kg
mass of the plane = -2500 kg (Negative mass doesn't make sense, so let's assume there was a typo in the question and the mass of the plane without fuel is actually 2500 kg.)
Next, let's calculate the change in momentum:
change in momentum = mass of the plane without fuel * velocity of the plane
To find the minimum velocity, we need to consider the vertical motion of the plane during takeoff. The upward force needed to take off is equal to the weight of the plane. So we can equate the change in momentum to the weight of the plane.
change in momentum = weight of the plane
mass of the plane without fuel * velocity of the plane = mass of the plane without fuel * g
velocity of the plane = g
Where g is the acceleration due to gravity, approximately 9.8 m/s^2.
Therefore, the minimum velocity of the exhaust gases required for the plane to take off is approximately 9.8 m/s.