A 1000kg rocket is set vertically on its launching pad. The propellent is expelled at the rate of 2kg/s. Find the minimum velocity of the exhaust gases so that rocket just begin to rise. Also find rocket's velocity 10second after ignition, assuming the minimum exhaust velocity.
Are you in calculus? The calculus problem here is that you have a constant pushing force (mg) however, the mass of the rocket is constantly decreasing mass=(1000-2t)g so acceleration is the inverse of that, which is not linear, and you have to integrate over the time 0 to 10 to find vfinal.
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To find the minimum velocity of the exhaust gases for the rocket to just begin to rise, we can use Newton's second law of motion, which states that the net force is equal to the product of mass and acceleration. The rocket will start to rise when the net force on it becomes positive.
Step 1: Calculate the weight of the rocket.
The weight of an object is given by the formula: weight = mass x acceleration due to gravity. The acceleration due to gravity is approximately 9.8 m/s^2.
weight = mass x acceleration due to gravity
weight = 1000 kg x 9.8 m/s^2
weight = 9800 N
Step 2: Calculate the force exerted by the expelled propellent.
The force exerted by the expelled propellent can be calculated using Newton's third law of motion, which states that for every action, there is an equal and opposite reaction. The force exerted by the expelled propellent is equal in magnitude but opposite in direction to the force pushing the rocket upward.
force of expelled propellent = mass flow rate x exhaust velocity
force of expelled propellent = 2 kg/s x v (m/s)
Step 3: Set up the equation to find the minimum exhaust velocity.
Since the rocket is just beginning to rise, the net force on the rocket is zero. Therefore, the weight of the rocket must be equal to the force exerted by the expelled propellent.
weight = force of expelled propellent
9800 N = 2 kg/s x v (m/s)
Step 4: Solve for the minimum exhaust velocity.
Divide both sides of the equation by 2 kg/s to isolate v.
v = 9800 N / 2 kg/s
v = 4900 m/s
Therefore, the minimum velocity of the exhaust gases is 4900 m/s.
To find the rocket's velocity 10 seconds after ignition, assuming the minimum exhaust velocity, we can use the concept of conservation of momentum.
Step 5: Calculate the impulse exerted by the expelled propellent.
Impulse is defined as the change in momentum of an object. The impulse exerted by the expelled propellent is equal to the rate of change of momentum, which can be calculated as force of expelled propellent x time.
impulse = force of expelled propellent x time
impulse = (2 kg/s x 4900 m/s) x 10 s
impulse = 98000 kg.m/s
Step 6: Calculate the change in momentum of the rocket.
The momentum of the rocket is equal to the product of mass and velocity.
initial momentum = mass x initial velocity
initial momentum = 1000 kg x 0 m/s
initial momentum = 0 kg.m/s
final momentum = mass x final velocity
final momentum = 1000 kg x v (m/s)
Change in momentum = final momentum - initial momentum
98000 kg.m/s = (1000 kg x v) - 0 kg.m/s
98000 kg.m/s = 1000 kg x v
Step 7: Solve for the rocket's velocity.
Divide both sides of the equation by 1000 kg to isolate v.
v = 98000 kg.m/s / 1000 kg
v = 98 m/s
Therefore, the rocket's velocity 10 seconds after ignition, assuming the minimum exhaust velocity, is 98 m/s.
To find the minimum velocity of the exhaust gases, we can use the concept of conservation of momentum. The momentum of a rocket is given by the equation:
Momentum = Mass × Velocity
Initially, the rocket is at rest on its launching pad, so its momentum is zero. Once the propellant is expelled, the rocket starts moving upward. At this point, we can assume that all of the initial momentum is given to the exhaust gases.
Let's say the minimum velocity of the exhaust gases is v. We can calculate the momentum of the exhaust gases by multiplying the mass expelled per second (2 kg/s) by the velocity v:
Momentum of exhaust gases = Mass expelled per second × Velocity of exhaust gases
Since the rocket is initially at rest, the momentum of the exhaust gases must be equal to the initial momentum of the rocket:
0 = 2 kg/s × v
Solving for v, we find:
v = 0 m/s
This means that the minimum velocity of the exhaust gases required for the rocket to just begin to rise is 0 m/s. In other words, the exhaust gases are expelled with no velocity.
Now, to find the rocket's velocity 10 seconds after ignition, assuming the minimum exhaust velocity, we can use Newton's Second Law of Motion:
Force = Mass × Acceleration
The force acting on the rocket is equal to the force generated by the expelled exhaust gases. The mass expelled per second is 2 kg/s, and for 10 seconds it will be 2 kg/s × 10 s = 20 kg.
The acceleration of the rocket can be calculated using the relationship between force, mass, and acceleration:
Force = Mass × Acceleration
Acceleration = Force / Mass
The force acting on the rocket is equal to the force generated by the expelled exhaust gases. The force can be calculated using the equation:
Force = Mass expelled per second × Velocity of exhaust gases
Since the minimum velocity of the exhaust gases is 0 m/s, the force acting on the rocket is also 0 N.
Acceleration = 0 N / (20 kg) = 0 m/s^2
This means that the rocket's acceleration is 0 m/s^2. Since acceleration is the rate of change of velocity, if the acceleration is 0, it implies that the velocity of the rocket remains constant.
Therefore, the rocket's velocity 10 seconds after ignition, assuming the minimum exhaust velocity, would still be 0 m/s.