In sinking a caisson in a muddy river bed the resistance is found to increase in direct proportion to the depth in the mud. A caisson weighing 6 tons sinks 4 feet under its own weight before coming to rest. Shew that, if a load of \(S\) tons is then suddenly added, it will sink 16 inches further.
Shew that, if \(AP=kBP\), where \(A\) and \(B\) are fixed points, the locus of \(P\) is a circle; and that for different values of \(k\) the different circles have the same radical axis. Shew that there are two circles of the system which touch a given straight line.
A shot is fired from a gun with velocity \(U\) and elevation \(\alpha\), so that it would hit an aeroplane at height \(h\) if the aeroplane were stationary. The aeroplane is, however, moving horizontally away from the gun with velocity \(V\); shew that it may nevertheless be hit if \[ (2U\cos\alpha-V)\sqrt{U^2\sin^2\alpha - 2gh} = UV\sin\alpha. \] \subsubsection*{Alternative questions in Physics.}
Find the magnitudes and directions of the axes of the conic \[ x^2+xy+y^2-2x+2y-6=0. \]
A balloon, whose capacity is 40,000 cubic feet, is filled with hydrogen, whose density is \(\cdot069\) that of air. Find the initial vertical acceleration when the temperature is 25\(^\circ\) C., if the total weight of envelope, car and passengers be 2500 lbs. The weight of a cubic foot of air at 0\(^\circ\) C. and at the pressure then prevailing is \(\cdot081\) lb.
Differentiate ab initio \(\text{cosec } x\), \(e^x\). Shew by differentiation that \[ \tan^{-1}\frac{1-x^2}{2x} + \sin^{-1}\frac{2x}{1+x^2} \] is a constant.
The melting point of lead is 333\(^\circ\) C., its specific heat is \(\cdot031\) and its latent heat of fusion 5.36. Find the least velocity in feet per second with which a lead bullet must strike a target into which it does not penetrate so that it may be melted, if the temperature of the bullet on striking the target be 150\(^\circ\) C. The work required to raise 1 lb. of water 1\(^\circ\) C. is 1400 ft. lbs.
Shew that, if \(\lambda\) is a repeated root of the equation \[ a_0\lambda^3+a_1\lambda^2+a_2\lambda+a_3=0, \] and if \(y=xe^{\lambda x}\), then \[ a_0\frac{d^3y}{dx^3} + a_1\frac{d^2y}{dx^2} + a_2\frac{dy}{dx} + a_3y=0. \]
Shew that the electrical resistance, measured between the opposite ends of a diagonal, of a framework consisting of 12 equal wires arranged along the edges of a cube is \(\frac{5}{6}\) of the resistance of one of the wires.
Find the shape of the circular cylinder, open at one end, which contains a maximum volume for a given superficial area.