4.II.9B

An octopus of mass $m_{o}$ swims horizontally in a straight line by jet propulsion. At time $t=0$ the octopus is at rest, and its internal cavity contains a mass $m_{w}$ of water (so that the mass of the octopus plus water is $m_{o}+m_{w}$ ). It then starts to move by ejecting the water backwards at a constant rate $Q$ units of mass per unit time and at a constant speed $V$ relative to itself. The speed of the octopus at time $t$ is $u(t)$, and the mass of the octopus plus remaining water is $m(t)$. The drag force exerted by the surrounding water on the octopus is $\alpha u^{2}$, where $\alpha$ is a positive constant.

Show that, during ejection of water, the equation of motion is

$m \frac{d u}{d t}=Q V-\alpha u^{2} .$

Once all the water has been ejected, at time $t=t_{c}$, the octopus has attained a velocity $u_{c}$. Use dimensional analysis to show that

$u_{c}=V f(\lambda, \mu)$

where $\lambda$ and $\mu$ are two dimensionless quantities and $f$ is an unknown function. Solve equation (1) to find an explicit expression for $u_{c}$, and verify that your answer is of the form given in equation (2).

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