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Rocket Mechanism

Find the mass...

Question

Find the mass $(m)$ of the rocket as a function of time, if it moves with a constant acceleration $a$ in the absence of external force. The gas escapes with a constant velocity $u$ relative to the rocket and its initial mass was $m_{0}$ .

m = m_{0} e^{- a t / u}

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m = m_{0} e^{- t / u}

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m = m_{0} e^{a t / u}

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m = m_{0} e^{2 a t / u}

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Solution

The correct option is A $m = m_{0} e^{- a t / u}$

Since $F_{e x t} = 0$ , we get
$F_{n e t} = V_{r} (\frac{- d m}{d t}) = u \frac{- d m}{d t}$ ... (I)
Also $F_{n e t} = m a$ ...(II)

By eq. (I) and (II)
$m a = - u \cdot \frac{d m}{d t}$
Integrating,
$\int_{m_{0}}^{m} \frac{d m}{m} = \int_{0}^{t} \frac{- a d t}{u}$
$\Rightarrow [ln \frac{m}{m_{0}}] = \frac{- a t}{u}$
$\Rightarrow \frac{m}{m_{0}} = e^{- a t / u}$
$\Rightarrow m = m_{0} e^{- a t / u}$

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