Calculate the force exerted on a rocket, given that the propelling gases are expelled at a rate of 1800 kg/s with a speed of 3.8×104 m/s (at the moment of takeoff).
To calculate the force exerted on the rocket, we can use Newton's third law of motion, which states that every action has an equal and opposite reaction.
According to this law, the force exerted by the rocket is equal in magnitude but opposite in direction to the force exerted by the expelled gases. Therefore, to calculate the force exerted on the rocket, we need to determine the momentum change of the expelled gases.
The momentum change can be calculated as follows:
Momentum Change = (mass of the gases) * (change in velocity of the gases)
Mass of the gases = 1800 kg/s
Change in velocity of the gases = 3.8×104 m/s (since the gases are expelled at this speed)
Now, let's calculate the momentum change:
Momentum Change = (1800 kg/s) * (3.8×104 m/s)
= 6.84×107 kg·m/s
Since the momentum change is equal to the impulse experienced by the rocket, we can now determine the force exerted on the rocket using the equation:
Force = Momentum Change / Time
Given that the momentum change occurs in a single second (as the gases are expelled at a constant rate of 1800 kg/s), the force exerted on the rocket is:
Force = (6.84×107 kg·m/s) / (1 s)
= 6.84×107 N
Therefore, the force exerted on the rocket is approximately 6.84×107 Newtons.
To calculate the force exerted on a rocket, we can use the principle of conservation of momentum. The momentum of the propelling gases being expelled from the rocket is equal to the momentum gained by the rocket in the opposite direction.
The formula for momentum is given by:
Momentum = mass × velocity
Let's assume the rocket has a mass of M kg and the velocity of the propelling gases is v m/s. The mass of the propelling gases expelled per second is 1800 kg/s, and its velocity is 3.8 × 10^4 m/s.
The momentum of the propelling gases expelled per second is:
Momentum of gases = mass × velocity
= (1800 kg/s) × (3.8 × 10^4 m/s)
Now, according to the principle of conservation of momentum, this momentum is equal to the momentum gained by the rocket:
Momentum of rocket = Mass of rocket × velocity of rocket
Since the initial velocity of the rocket is zero (at takeoff), we can simplify the equation:
Momentum of rocket = Mass of rocket × velocity of rocket
= M kg × 0 m/s
= 0
Therefore, the momentum gained by the rocket is zero.
Using the principle of conservation of momentum, we can equate the momentum of the propelling gases to the negative momentum of the rocket:
Momentum of gases = - Momentum of rocket
(1800 kg/s) × (3.8 × 10^4 m/s) = -M
Now, we can solve for the mass of the rocket:
M = - (1800 kg/s) × (3.8 × 10^4 m/s)
= - (6.84 × 10^7) kg-m/s
Since mass cannot be negative, the magnitude of the mass of the rocket is:
Mass of rocket = |M| = (6.84 × 10^7) kg-m/s
The force exerted on the rocket can then be calculated using Newton's second law of motion:
Force = mass × acceleration
At takeoff, the acceleration of the rocket can be determined using the equation:
Force = (1800 kg/s) × (3.8 × 10^4 m/s)
Therefore, the force exerted on the rocket is:
Force = (6.84 × 10^7) kg-m/s × (3.8 × 10^4 m/s)
force*time=mass*velocity
force=mass/time * velocity and of course, mass/time is mass rate, which is given as 1800kg/sec