An aircraft uses a rocket-assisted takeoff unit (RATO) that is capable of discharging 49.0 kg of gas in 15.0 s with an exhaust velocity of 1300 m/s. The 3000 kg aircraft, on an ordinary takeoff without RATO, requires a distance of 440 m to reach a lift-off speed of 210 km/hr with a constant propeller thrust. What is the thrust developed by the rocket?
.With both the RATO and the normal propeller thrust providing a constant acceleration, what is the distance required for lift-off?
To calculate the thrust developed by the rocket, we can use the principle of conservation of momentum.
The momentum change of the rocket can be calculated using the equation:
ΔP = (m * v) - (m₀ * v₀)
where:
ΔP is the change in momentum,
m is the mass of the gas discharged in the given time (49.0 kg),
v is the exhaust velocity of the gas (1300 m/s),
m₀ is the mass of the aircraft (3000 kg), and
v₀ is the initial velocity of the aircraft (0 m/s).
Substituting the given values into the equation, we have:
ΔP = (49.0 kg * 1300 m/s) - (3000 kg * 0 m/s)
= 63,700 kg·m/s
Since the change in momentum of the rocket is equal to the thrust developed, the thrust can be calculated as:
Thrust = ΔP / Δt
where Δt is the time taken for the gas to be discharged (15.0 s).
Substituting the values, we have:
Thrust = (63,700 kg·m/s) / (15.0 s)
= 4,246.67 N
Therefore, the thrust developed by the rocket is approximately 4,246.67 N.
For the second part of the question, where both the RATO and the normal propeller thrust provide a constant acceleration, we can use the equations of motion to calculate the distance required for lift-off.
The equation we can use is:
s = (v² - v₀²) / (2a)
where:
s is the distance required for lift-off,
v is the final velocity of the aircraft (converted to m/s from 210 km/hr),
v₀ is the initial velocity of the aircraft (0 m/s),
and a is the acceleration provided by both the RATO and propeller thrust.
First, convert the final velocity from km/hr to m/s:
v = (210 km/hr) * (1000 m/1 km) * (1 hr/3600 s)
≈ 58.33 m/s
Next, we need to calculate the acceleration. The net acceleration can be calculated using the equation:
a = ΣF / m
where:
ΣF is the sum of forces acting on the aircraft,
m is the mass of the aircraft.
Since there are two forces acting on the aircraft, the net force can be calculated as:
ΣF = Thrust + Propeller Thrust
where Thrust is the thrust developed by the rocket (4,246.67 N) and Propeller Thrust is the thrust required for the ordinary takeoff.
Since the situation involves a constant propeller thrust, we can calculate the propeller thrust using the equation:
Propeller Thrust = m * a
where m is the mass of the aircraft (3000 kg) and a is the constant acceleration.
Rearranging the equation, we have:
a = Propeller Thrust / m
Substituting the given values, we have:
a = (3000 kg * 58.33 m/s²) / 3000 kg
≈ 58.33 m/s²
Now we can substitute the values into the distance equation:
s = (58.33 m/s)² / (2 * 58.33 m/s²)
= 58.33 m
Therefore, the distance required for lift-off with both the RATO and the normal propeller thrust providing a constant acceleration is approximately 58.33 m.