Investigate the two dimensional motion of an incompressible fluid defined by the stream function \(\psi=A(x^2-y^2)\), and show that it can be produced from rest in the space between any two coaxial elliptic cylinders by rotating them with appropriate angular velocities.
Prove that when a gas flows in steady motion under the action of a pressure gradient only the velocity at a point where the pressure is \(p\) is \[ \left(2\int_p^{p_0} \frac{dp}{\rho}\right)^{\frac{1}{2}}, \] \(p_0\) being the pressure at points where the velocity is zero. Hence show that the maximum possible velocity which the gas can acquire is \[ C_m = \left(\frac{2\gamma p_0}{(\gamma-1)\rho_0}\right)^{\frac{1}{2}}, \] where \(\rho_0\) is the density corresponding with pressure \(p_0\). If the gas passes through a long straight tube in which the velocity is constant and equal to \(q\), show that the pressure at the nose of a projectile moving along the tube with velocity \(U\) in the opposite direction to the stream of gas is \[ p_0\left(1+\frac{2qU+U^2}{C_m^2}\right)^{\frac{\gamma}{\gamma-1}}. \]
Explain carefully how the azimuth of the sun at any given time at a known point on the earth's surface can be calculated from the data given in the Nautical Almanac. The altitude \(\alpha\) of the sun is observed at a place in latitude \(\phi\). If the sun's longitude is \(L\) and the obliquity of the ecliptic is \(\omega\), show that the azimuth \(A\) reckoned from the North is \[ \cos^{-1}\left(\frac{\sin L\sin\omega-\sin\alpha\sin\phi}{\cos\alpha\cos\phi}\right). \]
A cylindrical tin of negligible mass and made of very thin material contains some air and is held down in water by a vertical string attached to the middle point of its base, which is perforated. Show that it cannot be in unstable equilibrium with its axis vertical unless \(h\sqrt{2}