The first stage of a space vehicle consumed fuel and oxidizer at the rate of 1.65 104 kg/s with an exhaust speed of 2.55 103 m/s.
(a) Calculate the thrust produced by this engine.
(b) Find the acceleration the vehicle had just as it lifted off the launch pad on the Earth, taking the vehicle's initial mass as 3.00 106 kg.
To calculate the thrust produced by the engine, we can use the equation:
Thrust = Mass flow rate * Exhaust speed
(a) To find the thrust produced by the engine, we need to calculate the mass flow rate first. The mass flow rate is given as 1.65 × 10^4 kg/s.
Now we can calculate the thrust:
Thrust = 1.65 × 10^4 kg/s * 2.55 × 10^3 m/s
Thrust = 4.2075 × 10^7 N
Therefore, the thrust produced by this engine is 4.2075 × 10^7 N.
(b) To find the acceleration of the vehicle when it lifts off the launch pad, we can use Newton's second law of motion:
Force = mass * acceleration
In this case, the force is equal to the thrust produced by the engine, which we calculated in part (a).
Force = 4.2075 × 10^7 N
The mass of the vehicle is given as 3.00 × 10^6 kg.
Now we can calculate the acceleration:
4.2075 × 10^7 N = 3.00 × 10^6 kg * acceleration
acceleration = 4.2075 × 10^7 N / 3.00 × 10^6 kg
acceleration ≈ 14.03 m/s^2
Therefore, the acceleration of the vehicle just as it lifts off the launch pad on Earth is approximately 14.03 m/s^2.
To solve this problem, we can use the principle of conservation of momentum. The thrust produced by the engine can be calculated using the equation:
Thrust = (flow rate of propellant) * (exhaust speed)
(a) Calculating the thrust:
Given:
Flow rate of propellant (ṁ) = 1.65 x 10^4 kg/s
Exhaust speed (Ve) = 2.55 x 10^3 m/s
Thrust = (1.65 x 10^4 kg/s) * (2.55 x 10^3 m/s)
Thrust ≈ 4.21 x 10^7 N
Therefore, the thrust produced by the engine is approximately 4.21 x 10^7 N.
(b) Calculating the acceleration:
By Newton's second law, we know that the net force acting on the vehicle is equal to the product of its mass and acceleration:
Net force = mass * acceleration
In this case, the net force is equal to the thrust produced by the engine. The initial mass of the vehicle is given as 3.00 x 10^6 kg.
Thrust = mass * acceleration
4.21 x 10^7 N = 3.00 x 10^6 kg * acceleration
Therefore, the acceleration of the vehicle just as it lifted off the launch pad is:
acceleration = (4.21 x 10^7 N) / (3.00 x 10^6 kg)
acceleration ≈ 14 m/s^2
So, the acceleration of the vehicle just as it lifted off the launch pad on the Earth is approximately 14 m/s^2.