A rocket engine on a spacecraft of total mass 10,000kg ejected 1.4kg of hot gases every second at an average discontinuous speed of 240ms-1.
1- What was the force on the spacecraft?
2- What was the acceleration of the spacecraft?
3-If the engines were used for 30's, what was the impulse given to the spacecraft?
4- What was the change of velocity?
please help I've been thinking of how to solve it for an hour
so for change of momentum = 336 divided by 10,000? @Damon
To solve the given problem, we need to use the principles of Newton's third law of motion, which states that every action has an equal and opposite reaction. In this case, the ejection of hot gases from the rocket engine creates a reaction force on the spacecraft.
1. To find the force on the spacecraft, we can use the equation:
Force = Rate of change of momentum
We know that the rate of change of momentum is equal to the mass of gases ejected per second multiplied by the average speed of the gases.
Mass of gases ejected per second = 1.4 kg
Average speed of gases = 240 m/s
Therefore, the force on the spacecraft is:
Force = 1.4 kg/s * 240 m/s = 336 N
2. To find the acceleration of the spacecraft, we can use Newton's second law of motion, which states that force is equal to mass multiplied by acceleration:
Force = Mass * Acceleration
Rearranging the equation, we find:
Acceleration = Force / Mass
Mass of the spacecraft = 10,000 kg
Therefore, the acceleration of the spacecraft is:
Acceleration = 336 N / 10,000 kg = 0.0336 m/s^2
3. To find the impulse given to the spacecraft, we can use the impulse-momentum principle, which states that impulse is equal to the change in momentum:
Impulse = Change in momentum
The change in momentum can be calculated by multiplying the force by the duration of time that the force was applied. In this case, the engines were used for 30 seconds.
Therefore, the impulse given to the spacecraft is:
Impulse = Force * Time = 336 N * 30 s = 10,080 Ns
4. To find the change in velocity, we can use the equation:
Change in velocity = Impulse / Mass
Mass of the spacecraft = 10,000 kg
Impulse = 10,080 Ns
Therefore, the change in velocity of the spacecraft is:
Change in velocity = 10,080 Ns / 10,000 kg = 1.008 m/s
So, to summarize:
1. The force on the spacecraft is 336 N.
2. The acceleration of the spacecraft is 0.0336 m/s^2.
3. The impulse given to the spacecraft is 10,080 Ns.
4. The change in velocity of the spacecraft is 1.008 m/s.
To solve these questions, we can use the principle of conservation of momentum and the equations of motion. Here's how you can approach each problem:
1. What was the force on the spacecraft?
The force exerted by the rocket engine can be found using Newton's second law, which states that force (F) is equal to mass (m) multiplied by acceleration (a): F = m * a. In this case, the mass of the spacecraft is given as 10,000 kg, but we need to find the acceleration. The acceleration can be determined by dividing the change in momentum by the time interval during which this change occurred.
2. What was the acceleration of the spacecraft?
The acceleration of the spacecraft can be calculated by dividing the change in momentum by the time interval during which this change occurred. In this case, we know that the rocket engine ejected 1.4 kg every second. Since the mass of the ejected gases is known, we can calculate the change in momentum each second and then divide it by the mass of the spacecraft.
3. If the engines were used for 30 seconds, what was the impulse given to the spacecraft?
Impulse is defined as the change in momentum of an object and can be calculated by multiplying the force acting on the object by the time interval during which the force is applied. In this case, we need to calculate the force using the method described in the first question and then multiply it by the 30-second duration.
4. What was the change in velocity?
The change in velocity can be obtained by multiplying the acceleration of the spacecraft by the time interval during which this acceleration occurred. In other words, multiplying the acceleration by the time will give you the change in velocity.
Remember to use the appropriate units in each calculation and ensure that the given values are consistent with the units you are using for calculations.
force = rate of change of momentum
change of momentum of vehicle + change of momentum of fuel = 0
in one second:
10,000 (change of vehicle speed) = 1.4*240 = 336 kg m/s
so
change of vehicle momentum per second =336 kg m/s^2 = Force (answer #1)
acceleration = change of vehicle speed per second = 336*10^-4 =3.36*10^-2 m/s^2
impulse = change of momentum in 30 seconds = Force * 30
change of velocity = change of momentum / mass