The first stage of a space vehicle consumed fuel and oxidizer at the rate of 1.75 ✕ 104 kg/s, with an exhaust speed of 2.70 ✕ 103 m/s.
(a) Calculate the thrust produced by these engines.
N
(b) Find the acceleration of the vehicle just as it lifted off the launch pad on the Earth if the vehicle's initial mass was 3.00 ✕ 106 kg. [Note: You must include the force of gravity to solve part (b).]
m/s2 upward
Force up = rate of change of momentum
Per second momentum back = 1.75*2.7 * 10^7
= (4.73 *10^7 kg m/s)/s = 4.73*10^7 Newtons
F = m a
Fup - m g = m a
a = Fup/m - g
= 4.73*10^7/3*10^6 - 9.81
= 47.3/3 - 9.81
(a) Well, well, looks like we're dealing with rocket science here! To calculate the thrust produced by the engines, we can use a simple equation: thrust equals the exhaust speed times the rate at which fuel and oxidizer are consumed per second. So, thrust equals (2.70 ✕ 10^3 m/s) times (1.75 ✕ 10^4 kg/s). Crunching those numbers, we get a thrust of 4.73 ✕ 10^7 N. That's a whole lot of oomph!
(b) Ah, the good ol' acceleration on the launch pad. To solve this, we need to take into account two factors: the thrust provided by the engines and the force of gravity acting on the vehicle.
The net force acting on the rocket can be calculated by subtracting the force of gravity from the thrust. The force of gravity can be found using Newton's law of universal gravitation, which states that the force of gravity equals the mass of the rocket times the acceleration due to gravity (9.8 m/s^2 on Earth).
Net force = Thrust - Force of gravity
Net force = 4.73 ✕ 10^7 N - (3.00 ✕ 10^6 kg × 9.8 m/s^2)
Now, using Newton's second law (force equals mass times acceleration), we can find the acceleration:
Net force = mass × acceleration
4.73 ✕ 10^7 N - (3.00 ✕ 10^6 kg × 9.8 m/s^2) = (3.00 ✕ 10^6 kg) × acceleration
Now it's just a matter of solving for acceleration. Crunching the numbers, we get an upward acceleration of approximately 2.44 m/s^2. So, buckle up, because your space vehicle is about to take off at that acceleration!
To calculate the thrust produced by the engines, we can use the following formula:
Thrust (F) = (Mass Flow Rate) * (Exhaust Velocity)
Given:
Mass Flow Rate = 1.75 * 10^4 kg/s
Exhaust Velocity = 2.70 * 10^3 m/s
(a) Calculating the thrust:
F = (1.75 * 10^4 kg/s) * (2.70 * 10^3 m/s)
F ≈ 4.73 * 10^7 N
Therefore, the thrust produced by the engines is approximately 4.73 * 10^7 N.
To calculate the acceleration of the vehicle just as it lifted off the launch pad, we need to consider the force of gravity.
(b) Calculating the acceleration:
The net force acting on the vehicle will be the difference between the thrust force and the force due to gravity.
Force due to gravity is given by:
F_gravity = Mass * Acceleration due to gravity
Acceleration due to gravity on Earth is approximately 9.8 m/s^2.
Given:
Mass of the vehicle = 3.00 * 10^6 kg
F_gravity = (3.00 * 10^6 kg) * (9.8 m/s^2)
F_gravity ≈ 2.94 * 10^7 N
The net force is given by:
Net Force (F_net) = Thrust - F_gravity
F_net = (4.73 * 10^7 N) - (2.94 * 10^7 N)
F_net ≈ 1.79 * 10^7 N
Now we can use Newton's second law of motion to find the acceleration:
F_net = Mass * Acceleration
1.79 * 10^7 N = (3.00 * 10^6 kg) * Acceleration
Acceleration = (1.79 * 10^7 N) / (3.00 * 10^6 kg)
Acceleration ≈ 5.97 m/s^2 upward
Therefore, the acceleration of the vehicle just as it lifted off the launch pad on Earth is approximately 5.97 m/s^2 upward.
To calculate the thrust produced by the engines, we can use the equation:
Thrust = Flow rate * Exhaust speed
In this case, the flow rate is given as 1.75 ✕ 104 kg/s, and the exhaust speed is given as 2.70 ✕ 103 m/s.
(a) Calculate the thrust produced by the engines:
Thrust = 1.75 ✕ 104 kg/s * 2.70 ✕ 103 m/s
Thrust = 4.725 ✕ 107 kg⋅m/s
So, the thrust produced by the engines is 4.725 ✕ 107 N.
To calculate the acceleration of the vehicle just as it lifted off the launch pad on Earth, we need to consider both the thrust produced by the engines and the force of gravity acting on the vehicle.
The force of gravity can be calculated using the equation:
Force of gravity = Mass * Acceleration due to gravity
In this case, the mass is given as 3.00 ✕ 106 kg, and the acceleration due to gravity on Earth is approximately 9.8 m/s².
(b) Calculate the acceleration of the vehicle:
Force of gravity = 3.00 ✕ 106 kg * 9.8 m/s²
Force of gravity = 2.94 ✕ 107 N
The net force acting on the vehicle can be calculated as the difference between the thrust produced by the engines and the force of gravity:
Net force = Thrust - Force of gravity
Net force = 4.725 ✕ 107 N - 2.94 ✕ 107 N
Net force = 1.785 ✕ 107 N
Finally, we can use Newton's second law of motion, F = ma, to solve for the acceleration:
Net force = Mass * Acceleration
1.785 ✕ 107 N = 3.00 ✕ 106 kg * Acceleration
Acceleration = 1.785 ✕ 107 N / 3.00 ✕ 106 kg
Acceleration ≈ 5.95 m/s²
Therefore, the acceleration of the vehicle just as it lifted off the launch pad on Earth is approximately 5.95 m/s² upwards.