A ball whose coefficient of restitution is \(e\) is projected with velocity \(v\) at an inclination \(\alpha\) to the horizontal from a point \(A\) on a horizontal plane. \(A\) is at a distance \(d\) from a vertical wall. The ball strikes the wall, and then after rebounding once on the horizontal plane returns to \(A\). Prove that \[ v^2e\sin2\alpha=gd. \]
Two particles, each of mass \(m\), are attached to the ends of a long fine inextensible string, which hangs over two small smooth pegs which are at the same level and \(2a\) apart. A particle of mass \(2m\) is attached to the string midway between the pegs and is then let go. Prove that during the subsequent motion, if \(\phi\) is the angle between the two non-parallel parts of the string, the velocity of the mass \(2m\) is \[ 2\sqrt{ag\frac{1-\tan\frac{\phi}{4}}{3+\cos\phi}}. \]
A uniform rectangular plate \(ABCD\) is hinged at the fixed point \(A\) and is supported in such a position that \(AB\), one of the longer sides, is horizontal, and \(AD\) is vertical. When the plate is released it swings in its own plane about the fixed hinge \(A\) and comes to rest with \(AB\) vertical. The stiffness of the hinge produces a constant retarding couple during motion. Prove that the plate stays in the new position if \[ \frac{AB}{AD} > 1+\frac{\pi}{2}. \]
The acceleration of a certain racing motor car at a speed of \(v\) feet per second is \(\left(3.6 - \frac{v^2}{9000}\right)\) feet per second per second. Find the maximum speed of the car, and prove that from a standing start a speed of 150 feet per second is acquired in one minute after travelling 1800 yards. Assume that \(\log_e 6=1.8\), and \(\log_e 11 = 2.4\).
If \[ y = \frac{\sin^{-1}x}{\sqrt{1-x^2}}, \] prove that \[ (1-x^2)\frac{dy}{dx} = xy+1; \] and if \(y_n\) denotes the \(n\)th differential coefficient of \(y\), prove that, when \(x=0\), \[ y_n=(n-1)^2y_{n-2}. \] Prove that the limit of \((\cos x)^{\cot^2x}\) as \(x\) tends to zero is \(\displaystyle\frac{1}{\sqrt{e}}\).
If \(\alpha\) and \(\beta\) are given acute angles, and \(\alpha>\beta\), prove that the maximum and minimum values of \[ \frac{1+2x\cos\alpha+x^2}{1+2x\cos\beta+x^2} \] are \[ \frac{1-\cos\alpha}{1-\cos\beta} \quad \text{and} \quad \frac{1+\cos\alpha}{1+\cos\beta} \text{ respectively}. \]
Sketch the locus of a point \(P\) for which \[ x=a\cos^3\phi, \quad y=a\sin^3\phi, \] where \(a\) is constant and \(\phi\) is variable. Prove that the tangent at \(P\) to the locus is \[ x\sec\phi+y\csc\phi=a, \] and that the whole length of the curve is \(6a\).
Two equal ladders are hinged at the top and rest on a rough floor forming an isosceles triangle with the floor of vertical angle \(2\theta\). A man whose weight is \(n\) times that of either ladder goes slowly up one of them. Calculate the reactions at the floor when his distance from the top is \(x\), and show that slipping begins when \[ nx/l=(2\mu-\tan\theta)/(\mu-\tan\theta). \]
A frame, formed of four light rods of equal length, freely jointed at \(A,B,C,D\), is suspended at \(A\). A particle of mass \(m\) is suspended from \(B\) and \(D\) by two strings each of length \(l\). The frame is prevented from collapsing by a string \(AC\). Show that the tension of the string is equal to \(\frac{1}{2}mg\frac{AP}{PN}\), where \(P\) is the particle and \(N\) is the centre of the rhombus \(ABCD\).
\(AB\) is the horizontal diameter of a circular wire whose plane is vertical. A bead of mass \(M\) at the lowest point \(C\) can slide on the wire and is attached to two strings which pass through small fixed rings at \(A,B\). To the other ends of the strings are attached equal particles \(m\) which hang freely. Find the potential energy of the system when it is displaced so that the radius to \(M\) makes an angle \(\theta\) with the vertical. Deduce that the equilibrium with \(M\) at \(C\) is stable if \(m