Show that for an open system such as a rocket, the second rocket equation is
Answer to Problem 83PQ
It is shown that for an open system the second rocket equation is
Explanation of Solution
Consider the closed rocket exhausted fuel system and imagine an external force acting on the system.
Write the expression for the Newton’s second law.
Here,
Write the expression for the total momentum.
Here,
Use equation (II) in equation (I),
The change in momentum of the exhaust is the same in both frame of reference of the Earth and the rocket so that,
Use equation (IV) in equation (III),
Write the expression for the momentum of an moving object.
Here,
When a small amount of mass
The mass in the exhaust leaves the system so that the mass exhausted will equal the mass lost by the rocket so that,
Use equation (VIII) in equation (VII),
Write the expression for the force on the rocket in the Earths frame of reference.
The acceleration on the rocket in the Earths frame of reference is given by,
Use equation (XI) in (X),
Write the expression for the force exerted by the fuel which is known as thrust,
Use equation (XII) in equation (X), and use (XIII) in equation (IX)
Rearrange the above equation,
Conclusion:
Therefore, it is shown from equation (XIV) that for an open system the second rocket equation is
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