The first stage of a space vehicle consumed fuel and oxidizer at the rate of 1.60 multiplied by 104 kg/s, with an exhaust speed of 2.55 multiplied by 103 m/s.
(a) Calculate the thrust produced by these engines.
answer should be in 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 multiplied by 106 kg. [Note: You must include the force of gravity to solve part (b).]
answer should be m/s2 upward
Thrust= mass*v/time= mass/time*v
b)Thrust-mg=ma solve for a
To solve this problem, we need to use Newton's second law of motion: force (thrust) is equal to the mass multiplied by the acceleration.
(a) To calculate the thrust produced by the engines, we will use the formula:
Thrust = (Fuel consumption rate) x (Exhaust speed)
Given:
Fuel consumption rate = 1.60 x 10^4 kg/s
Exhaust speed = 2.55 x 10^3 m/s
Thrust = (1.60 x 10^4 kg/s) x (2.55 x 10^3 m/s)
Multiply the numbers:
Thrust = 4.08 x 10^7 kg.m/s²
Thrust = 4.08 x 10^7 N
Therefore, the thrust produced by the engines is 4.08 x 10^7 Newtons.
(b) To find the acceleration of the vehicle just as it lifted off the launch pad on Earth, we need to consider the force of gravity acting on the vehicle.
The net force acting on the vehicle is the difference between the force of thrust and the force of gravity.
Force of gravity = mass x acceleration due to gravity
Given:
Mass of the vehicle = 3.00 x 10^6 kg
Acceleration due to gravity on Earth (g) = 9.8 m/s²
Force of gravity = (3.00 x 10^6 kg) x (9.8 m/s²)
Force of gravity = 2.94 x 10^7 kg.m/s²
Force of gravity = 2.94 x 10^7 N
Net force = Thrust - Force of gravity
Net force = (4.08 x 10^7 N) - (2.94 x 10^7 N)
Perform the subtraction:
Net force = 1.14 x 10^7 N
Now, we can use Newton's second law to find the acceleration:
Net force = mass x acceleration
1.14 x 10^7 N = (3.00 x 10^6 kg) x acceleration
Divide both sides by the mass:
acceleration = (1.14 x 10^7 N) / (3.00 x 10^6 kg)
acceleration = 3.8 m/s²
Therefore, the acceleration of the vehicle just as it lifted off the launch pad on Earth is 3.8 m/s² upward.