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The time rate of change of the position of an object
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How can kinetic energy be proportional to the square of velocity, when velocity is relative?
The velocity of the exhaust $V_e$ in the observer frame is related to the velocity of the exhaust in the rocket frame $v_e$ by (since exhaust velocity is in the negative direction)
$$V_e=V_{rocket}-v_e … Which tells us that there is a cloud of dust or fuel going the other direction with a speed of $100V$
now we check the kinetic energy used to reach that velocity.
$$ E_2 = \frac{1}{2} \left[ m_{final} …