Dynamics | Part IA, 2008

An octopus of mass mom_{o} swims horizontally in a straight line by jet propulsion. At time t=0t=0 the octopus is at rest, and its internal cavity contains a mass mwm_{w} of water (so that the mass of the octopus plus water is mo+mwm_{o}+m_{w} ). It then starts to move by ejecting the water backwards at a constant rate QQ units of mass per unit time and at a constant speed VV relative to itself. The speed of the octopus at time tt is u(t)u(t), and the mass of the octopus plus remaining water is m(t)m(t). The drag force exerted by the surrounding water on the octopus is αu2\alpha u^{2}, where α\alpha is a positive constant.

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

mdudt=QVαu2.m \frac{d u}{d t}=Q V-\alpha u^{2} .

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

uc=Vf(λ,μ)u_{c}=V f(\lambda, \mu)

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

Typos? Please submit corrections to this page on GitHub.