Rocket Propulsion Gas Injection

[ada0289]

Please help, book gives no examples relatively close to this.

A rocket is fired in deep space, where gravity is negligible.


If the rocket has an initial mass of 6000 kg and ejects gas at a relative velocity of magnitude 2000 m/s , how much gas must it eject in the first second to have an initial acceleration of 25.0 m/s^2?

 

[tiny-tim]

Welcome to PF!

Hi ada0289! Welcome to PF!

If the rocket has an initial mass of 6000 kg and ejects gas at a relative velocity of magnitude 2000 m/s , how much gas must it eject in the first second to have an initial acceleration of 25.0 m/s^2?

Hint: use conservation of momentum

What do you get?

 

[nlightner]

To determine the amount of gas that must be ejected in the first second for the rocket to have an initial acceleration of 25.0 m/s^2, we can use the equation F = ma, where F is the force, m is the mass, and a is the acceleration.

Since we know the initial mass of the rocket (6000 kg) and the desired initial acceleration (25.0 m/s^2), we can rearrange the equation to solve for the force: F = ma = (6000 kg)(25.0 m/s^2) = 150,000 N.

Now, to calculate the force generated by the gas ejection, we can use the equation F = Δp/Δt, where Δp is the change in momentum and Δt is the time interval.

Since the rocket ejects gas at a relative velocity of 2000 m/s, the change in momentum can be calculated as Δp = (6000 kg)(2000 m/s) = 12,000,000 kg·m/s. Assuming the gas is ejected in the first second, the time interval Δt would be 1 second.

Plugging these values into the equation, we get: F = Δp/Δt = (12,000,000 kg·m/s) / (1 s) = 12,000,000 N.

Therefore, in order for the rocket to have an initial acceleration of 25.0 m/s^2, it would need to eject gas with a total force of 12,000,000 N in the first second. This amount of gas can then be converted into a mass using the equation F = ma, where F is the force calculated (12,000,000 N), m is the mass of the gas, and a is the acceleration of the gas (2000 m/s^2).

Thus, the amount of gas ejected in the first second would be approximately 6 kg. This calculation shows that even a small amount of gas ejected at a high velocity can generate a significant amount of force and contribute to the rocket's acceleration.