n 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?

Question

n 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?

Answer 1

thrustrocket=massgas/time * velocitygas

= 49kg/15sec*1300m/s=4247N

Answer 2

To calculate the thrust developed by the rocket, we need to use Newton's second law, which states that force (thrust) is equal to mass times acceleration. In this case, the acceleration is the change in velocity over time.

First, let's calculate the change in velocity of the aircraft during takeoff without RATO. We are given that the aircraft reaches a lift-off speed of 210 km/hr, which we need to convert to m/s.

1 km/hr = (1 km / 3600 s) = (1000 m / 3600 s) = (5/18) m/s
210 km/hr = (210 * 5/18) m/s ≈ 58.33 m/s

Next, we need to calculate the acceleration of the aircraft during takeoff without RATO. We can use the equation: acceleration = change in velocity / time taken.

We are given that the aircraft covers a distance of 440 m during takeoff. To find the time taken, we can use the formula: distance = (1/2) * acceleration * time^2.

Rearranging the formula, we have: time^2 = (2 * distance) / acceleration.
Substituting the values, we get: time^2 = (2 * 440) / acceleration.

Simplifying, we have: time^2 = 880 / acceleration.

To solve for acceleration, we need to find the square root of both sides: acceleration = √(880 / time^2).

Now, we can calculate the time taken using the equation: time = (change in velocity) / acceleration.
Substituting the values, we get: time = 58.33 / acceleration.

Since we want to know the thrust developed by the rocket, not the time, we'll keep the variable "time" in the equation.

Now, let's calculate the thrust developed by the rocket:
Force = mass * acceleration.

We know that the mass of the gas discharged by the rocket is 49.0 kg and the exhaust velocity is 1300 m/s. Using the equation for acceleration, we have:

acceleration = (change in velocity) / (time taken) = (1300 m/s) / (15.0 s).

Finally, substituting the values into the equation for force, we get:

Force = mass * acceleration = (49.0 kg) * (1300 m/s) / (15.0 s).

Evaluating this expression, we can find the thrust developed by the rocket.