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
Obtain the equation \(y=c\cosh\frac{x}{c}\) for the curve of a uniform chain hanging under gravity. If the chain is suspended from two points \(A,B\) on the same level and the depth of the middle below \(AB\) is \(l/n\), where \(2l\) is the length of the chain, show that the horizontal span \(AB\) is equal to \(l\left(n-\frac{1}{n}\right)\log\left(\frac{n+1}{n-1}\right)\). Find an approximation to the difference between the arc \(AB\) and the span \(AB\) when \(n\) is large.
A load \(W\) is to be raised by a rope, from rest to rest, through a height \(h\); the greatest tension which the rope can safely bear is \(nW\). Show that the least time in which the ascent can be done is \(\left\{\frac{2nh}{(n-1)g}\right\}^{\frac{1}{2}}\).
Find the horse power required to lift 1000 gallons of water per minute from a canal 20 feet below and project it from a nozzle of cross section 2 sq. inches. [1 c. foot of water weighs 62\(\frac{1}{2}\) lb. and 1 gallon of water weighs 10 lb.]