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A rocket accelerates straight up by ejec...

A rocket accelerates straight up by ejecting gas downwards. In a small time interval `Deltat`, it ejects a gas of mass `Delta m` at a relative speed `u` . Calculate KE of the entire system at `t+Deltat` and `t` and show that the device that ejects gas does work `=((1)/(2))Delta m . u^(2)` in this time interval (neglect gavity).

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To solve the problem, we need to calculate the kinetic energy of the entire system at two different times and show that the work done by the device that ejects gas is equal to \(\frac{1}{2} \Delta m \cdot u^2\). ### Step-by-Step Solution 1. **Define Initial Conditions**: At time \(t\), the rocket has a mass \(m\) and a velocity \(v\). The kinetic energy \(E_1\) at this time can be expressed as: \[ E_1 = \frac{1}{2} m v^2 ...
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