A jumbo jet airplane has a mass of about 400,000 kg when fully loaded with over
400 passengers and takes off at a speed of 250 km/h. Determine;The takeoff speed when the airplane has 100 empty seats. Assume each passenger with luggage is 140 kg and the wing and flap settings are
maintained the same.
Lift == k v^2
Lift at 250 km/h = 400,000 g
100 * 140 = 14,000 kg
so lift needed = (400,000 -14,000) g Newtons
= 386 g Newtons
250^2/400,000 g = v^2 / 386,000 g
v^2 = 250^2 (386/400)
v = 250 sqrt (.965)
v = 245.6 km/h
A 40 tonne jumbo jet accelerates down the runway at 5 m/s^2. What must be the thrust on each of its four engines? What is the horizontal force that a 700 N pilot exerts on his seat?
To determine the takeoff speed when the airplane has 100 empty seats, we need to adjust the total mass of the airplane by subtracting the mass of the empty seats.
1. Calculate the mass of the loaded airplane:
Number of passengers = 400
Mass of each passenger with luggage = 140 kg
Total mass of passengers = Number of passengers * Mass of each passenger
= 400 * 140 kg
= 56,000 kg
Total mass of the loaded airplane = Mass of loaded airplane - Mass of passengers
= 400,000 kg - 56,000 kg
= 344,000 kg
2. Calculate the mass of the airplane when it has 100 empty seats:
Number of empty seats = 100
Mass of each empty seat = Mass of each passenger with luggage = 140 kg
Total mass of empty seats = Number of empty seats * Mass of each empty seat
= 100 * 140 kg
= 14,000 kg
Total mass of the airplane with 100 empty seats = Total mass of the loaded airplane - Total mass of empty seats
= 344,000 kg - 14,000 kg
= 330,000 kg
3. Calculate the takeoff speed with 100 empty seats.
Mass of the airplane with 100 empty seats = 330,000 kg
Speed of the airplane with 100 empty seats = Speed of the fully loaded airplane * √(Mass of fully loaded airplane / Mass of the airplane with 100 empty seats)
= 250 km/h * √(400,000 kg / 330,000 kg)
= 250 km/h * √(1.2121)
= 250 km/h * 1.10
= 275 km/h
Therefore, the takeoff speed when the airplane has 100 empty seats is approximately 275 km/h.
To determine the takeoff speed when the airplane has 100 empty seats, we need to calculate the change in mass of the airplane.
Step 1: Calculate the mass of the passengers and their luggage.
Each passenger with luggage weighs 140 kg, so the total weight of the passengers and their luggage is:
400 passengers * 140 kg/passenger = 56,000 kg
Step 2: Calculate the current mass of the fully loaded airplane.
The given mass of the fully loaded airplane is 400,000 kg, so we subtract the mass of the passengers and their luggage to find the mass of the airplane itself:
Mass of airplane = 400,000 kg - 56,000 kg = 344,000 kg
Step 3: Calculate the mass of the airplane with 100 empty seats.
Since each passenger with luggage weighs 140 kg, 100 empty seats represent a reduction in mass of:
100 empty seats * 140 kg/passenger = 14,000 kg
The mass of the airplane with 100 empty seats is:
Mass of airplane with 100 empty seats = Mass of airplane - Mass reduction due to empty seats
Mass of airplane with 100 empty seats = 344,000 kg - 14,000 kg = 330,000 kg
Step 4: Calculate the takeoff speed with 100 empty seats.
To calculate the takeoff speed, we use the principle of conservation of linear momentum.
The momentum before and after takeoff remains the same.
Momentum before takeoff = Momentum after takeoff
P(before) = P(after)
Since momentum is given by the product of mass and velocity, we have:
Mass_before * Velocity_before = Mass_after * Velocity_after
Solving for Velocity_after (takeoff speed with 100 empty seats):
Velocity_after = (Mass_before * Velocity_before) / Mass_after
Plugging in the values:
Velocity_before = 250 km/h = 250,000 m/h (converted to meters per hour)
Mass_before = 400,000 kg (fully loaded airplane)
Mass_after = 330,000 kg (airplane with 100 empty seats)
Velocity_after = (400,000 kg * 250,000 m/h) / 330,000 kg = 303,030 m/h
Therefore, the takeoff speed when the airplane has 100 empty seats is approximately 303,030 m/h.