A particle is projected in a fixed vertical plane from a point \(O\) with velocity \(\sqrt{2ga}\) and the upward vertical component is \(v\). Show that after time \(2a/v\) the particle is on a fixed parabola independent of the value of \(v\). Show also that the actual path touches the fixed parabola at this point and that the direction of motion there is perpendicular to the direction of projection.
Find the radial and transverse components of the acceleration of a point moving in a plane and whose position at any time is described by polar coordinates \(r, \theta\). A fine string is being unwound from a flat circular reel and the free portion \(PT\) is kept taut and in the plane of the reel. Show that if the point of contact \(T\) moves with constant angular velocity round the reel, the acceleration of the end \(P\) is always towards a certain point on the (moving) radius through \(T\), and determine its magnitude.
A heavy tube \(ABC\) is bent at right angles at \(B\) and the part \(AB\) is horizontal and slides freely through two fixed rings while the part \(BC\) is vertical. Two particles \(P\) and \(Q\) each of mass equal to that of the tube move in \(AB\) and \(BC\) respectively and are connected by a light inextensible string that can slide freely on the inside of the tube. If the system is released from rest, find the velocity of \(Q\) when it has descended a distance \(y\). Show also that the horizontal and vertical components of acceleration of \(Q\) are in the ratio 1:3.
A heavy flywheel which is known to be rotating with average angular velocity \(p\) is being driven by a variable driving couple \(G\sin^2 pt\) and retarded by a constant torque \(\frac{1}{2}G\) opposing the motion. Find the least moment of inertia that the flywheel must possess in order that the difference between its greatest and least angular velocities shall be less than \(p/N\), where \(N\) is a given large number.
A particle moves under an attraction varying inversely as the square of the distance from a fixed centre, and is describing a circle with period \(T\). Show that, if it is suddenly stopped and then allowed to fall freely, it will reach the centre of force after a time \(T/4\sqrt{2}\).
A uniform straight tube of mass \(M\) rests freely on a smooth horizontal table and contains a particle of mass \(m\) at rest at its mid-point. The system is set rotating about a vertical axis through its centre with angular velocity \(\omega\). Show that if the particle is slightly disturbed it will eventually leave the tube, and that at the instant it does so the angular velocity of the system will be \[ \frac{M+m}{M+4m}\omega. \]
Show that the simultaneous equations \begin{align*} ax+y+z&=p, \\ x+ay+z&=q, \\ x+y+az&=r \end{align*} have an unique solution if \(a\) has neither of the values 1 or \(-2\). Show also that, if \(a = -2\), there is no solution unless \(p, q\) and \(r\) satisfy a certain condition, and that there are then an infinite number of solutions. Discuss the solution of the equations when \(a=1\). Find the most general solution (if any) in the following cases: (i) \(a=3, p=q=r=1\), (ii) \(a=-2, p=q=r=1\), (iii) \(a=-2, p=1, q=-1, r=0\), (iv) \(a=1, p=q=r=0\).
If \(\alpha\) is a complex root of the equation \(x^7-1=0\), express the other six roots in terms of \(\alpha\). Show that \(\alpha+\alpha^2+\alpha^4\) is a root of a quadratic equation whose coefficients do not involve \(\alpha\). Prove that \[ \cos\frac{\pi}{7} - \cos\frac{2\pi}{7} + \cos\frac{3\pi}{7} = \frac{1}{2}, \quad -\sin\frac{\pi}{7} + \sin\frac{2\pi}{7} + \sin\frac{3\pi}{7} = \frac{\sqrt{7}}{2}. \]
Define the function \(e^y\), and deduce from your definition that, for all values of \(n\), \(y^n e^{-y} \to 0\) as \(y\to\infty\). Examine the behaviour of the following functions as \(x\) varies through real values, and in particular discuss their gradients for small positive and negative values of \(x\). Illustrate your results by sketch-graphs. \[ \text{(i) } \tanh\frac{1}{x}, \quad \text{(ii) } x\tanh\frac{1}{x}. \]
\(f(x)\) is a continuous function with continuous first, second and third derivatives, and \[ R(x) = \frac{1}{2} \int_0^x (x-t)^2 f'''(t) \,dt. \] Prove by integration by parts that \[ f(x) = f(0) + xf'(0) + \frac{x^2}{2}f''(0) + R(x). \] Show also that \[ R(x) = \frac{x^3}{3!}f'''(\theta x), \] where \(0 < \theta < 1\). State and prove a more general result applicable to a function with continuous derivatives up to and including the \(n\)th.