Problems

A particle is projected in a given vertical plane from a point \(O\), the horizontal and vertical components of velocity being \(u\) and \(v\) respectively. If \(u, v\) are connected by a relation \[ au^2 + v^2 = 2gh, \] where \(a, h\) are positive constants, shew that the envelope of the trajectories is a parabola whose vertex is at a height \(h\) above \(O\) and whose latus rectum is \(4h/a\). Shew also that to reach the point \(Q\) on the envelope the elevation of the direction of projection must be \(\tan^{-1}(2h/x)\), where \(x\) denotes the horizontal projection of \(OQ\). A gun is mounted on a truck and can fire a shell of mass \(m\) in a vertical plane parallel to the rails, the mass of the gun and truck together being \(M\). Find the envelope of the trajectories (i) on the assumption that the velocity of the shell relative to the gun is constant and equal to \(\sqrt{2gh}\), (ii) on the assumption that the total kinetic energy immediately after the shell leaves the gun is constant and equal to \(mgh\).

Explain briefly the principles of conservation of momentum and energy, and apply them to the solution of the following problem. A bead of mass \(M\) slides on a smooth straight horizontal wire, and a light rod of length \(l\) is freely attached to the bead and carries a particle of mass \(m\) at the other end. The rod is held in the vertical plane through the wire at an inclination \(\alpha\) to the vertical, and released. Shew that, if the inclination of the rod to the vertical at a subsequent time is \(\theta\), then \[ (M+m\sin^2\theta)l\dot{\theta}^2 = 2(M+m)g(\cos\theta - \cos\alpha). \]

If \(\alpha, \beta, \gamma\) be the distances of the centre of the nine-point circle from the vertices of a triangle, \(p\) its distance from the orthocentre, and \(R\) the radius of the circumscribing circle, then \[ \alpha^2+\beta^2+\gamma^2+p^2 = 3R^2. \]

Show that the equation \[ \{a(x^2-c)-b(y^2-c)\}^2 + 4\{axy+h(y^2-c)\}\{bxy+h(x^2-c)\}=0 \] represents four lines forming the sides of a rhombus.

A rectangular plate has sides ten inches and five inches. If equal squares are cut out at the four corners and if the sides are then turned up so as to form an open box, find the volume of the greatest box so formed.

Trace the curve \[ x = 2a \sin^2 t \cos 2t, \quad y = 2a \sin^2 t \sin 2t. \] Show that the length of its arc is \(8a\), and find the radius of curvature at the origin.

Show that if \[ I_n = \int_0^\infty \frac{dx}{(1+x^2)^n}, \] where \(n\) is a positive integer, then \[ I_{n+1} = \left(1-\frac{1}{2n}\right)I_n. \] and hence evaluate \(I_n\).

Prove that \[ 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n-1}-\log n \] tends to a finite limit, as \(n\to\infty\). If \(p=2^q\), where \(q\) is an integer, prove that \[ \sum_{r=2}^p \frac{1}{r \log r} > \frac{\log(q+1)}{2 \log 2}. \]

Two strings, each of length \(l\), are attached to a ceiling, and the lower ends are attached to a magnet of moment \(M\), length \(l\), and weight \(W\). When the strings are vertical the magnet is in the magnetic meridian but with its north-seeking pole towards the south. Through what angle will it have to turn before it comes to another position of equilibrium? (Assume that the earth's magnetic field exists a couple \(HM \sin\theta\) on the magnet when it makes an angle \(\theta\) with the magnetic meridian.)

The case of a rocket weighs 2 lbs. and the charge 5 lbs. The charge burns at a uniform rate and is completely burnt in 3 seconds, during which time it exerts a constant propulsive force of 20 lb.-wt. If the rocket is fired vertically, find the vertical velocity acquired during the burning of the charge.