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A rocket is fired in deep space, where gravity is negligible. If the rocket has an initial mass of 6000 $\mathrm{kg}$ and ejects gas at a relative velocity of magnitude 2000 $\mathrm{m} / \mathrm{s}$ , how much gas must it eject in the first second to have an initial acceleration of 25.0 $\mathrm{m} / \mathrm{s}^{2}$ .

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Physics 101 Mechanics

Chapter 8

Momentum

Physics Basics

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

University of Washington

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in this problem. We're told that a rocket with an initial mass of 6000 kilograms ejects gas behind it with a velocity with positive being the rockets velocity eso The gas has a velocity of negative 2000 meters per second. And, um, we wants the rocket to achieve an acceleration of 25.0 meters per second squared, ejecting the gas for one second. So we need to find out how much mass must be ejected. So how much mass of the guests that's being exhausted at the back? Um, it's necessary to achieve this, um, this acceleration over this amount of time. So, um, we want to start with our familiar rocket equation. So, um, the force of threats which is equal to the mass of the rocket times thea Acceleration of the rocket, um, is equal to, um, negative velocity of the exhaust, um, times the rate at which mass is ejected. So we just want to re arrange for the mass here, Um, so Delta M. Or we want to re arrange for them mass loss here. So don't m equals the initial mess. That rocket times the acceleration of the rocket times um the exhaust philosophy and this is negative. Um, are sorry times the time over which the gases ejected, divided by the exhaust velocity. No, we have all of these constants. They are given to us. So just plugging them in, we have that the, um, amount of mass that must be rejected is equal to negative 6000 kilograms times the acceleration, which is 25.0 meters per second squared times. Um, Delta T, which is 1.0 seconds. This is all divided by the speed of the exhaust, which is of course, negative 2000 meters per second. Now plugging that all into a calculator, we find that the amount that of gas that it must ejects to achieve this acceleration is 75 kilograms. Therefore, 75 0.0 kilograms of gas must be ejected.

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