Problems

An anti-tank gun fires a projectile weighing 2 lb. with a muzzle velocity of 3000 ft. per sec. The shell travels 9 ft. before leaving the muzzle, and the rate of burning of the charge is such that the acceleration of the shell is sensibly uniform throughout its passage to the muzzle. Show that the (sensibly constant) force exerted on the base of the shell by the gases is approximately \(3 \times 10^4\) pounds weight. \par The gun itself weighs 2 cwt. and is checked in its recoil by springs which allow a recoil of 1 ft. Show that at the fullest point of recoil the force exerted by the springs is approximately \(5 \times 10^3\) pounds weight. (It may be assumed that the gun does not recoil appreciably while the shell is in the muzzle.)

Neglecting air resistance, show that, for a projectile fired under gravity, the maximum range on a horizontal plane is \(V^2/g\), where \(V\) is the muzzle velocity and \(g\) is the acceleration of a freely falling body. For a 2-in. mortar whose maximum range is 530 yd., show that the muzzle velocity is approximately 225 ft. per sec. Given that \(\tan 36^\circ 52' = \frac{3}{4}\), show that the range corresponding to an elevation of \(36^\circ 52'\) is 509 yd. What other elevation gives the same range? \par An approximate allowance for the effect of a head wind of velocity \(w\) is given by assuming that the horizontal component of the velocity of the projectile suffers a constant retardation \(kw\). Two successive shots are fired against a head wind from the mortar at the two (low angle and high angle) elevations which in still air would give a range of 509 yd. If \(w\) is such that \(kw = 2 \text{ ft./(sec.)}^2\), show that the actual ranges are 485 yd. and 467 yd.

A particle of mass \(m\) is constrained to move on a smooth wire in the shape of a parabola whose axis is vertical and whose vertex is upwards. The particle is projected from the vertex with velocity \(u\). Show that the pressure on the wire at any point is \[ \frac{m}{\rho}(gl-u^2), \] where \(2l\) is the latus rectum, and \(\rho\) is the radius of curvature.

A uniform rod of mass \(m\) and length \(2a\) is supported horizontally by two elastic strings, each of natural length \(l\) and modulus of elasticity \(\lambda\), which are attached to a fixed point vertically above the middle of the rod. In equilibrium the strings make an angle \(\theta\) with the vertical. Show that the period of small oscillations in which the rod remains horizontal is \(2\pi/n\), where \[ n^2 = \frac{g}{a}\frac{a-l\sin^3\theta}{a-l\sin\theta}\tan\theta. \]

A uniform rod of mass \(m\) and length \(2a\) is inclined at an angle \(\theta\) to the vertical and falls without rotation so as to impinge with velocity \(u\) on a perfectly rough inelastic horizontal plane. Determine the impulsive force on the rod. \par Show also that when the rod becomes horizontal the impulse imparted to the plane is \(\frac{2}{3}m\sqrt{(u^2\sin^2\theta + \frac{3}{4}ag\cos\theta)}\).

Show how to perform any three of the following constructions, using a ruler only. Justify your constructions.

1. [(i)] Given three collinear points \(A, B, C\), find the harmonic conjugate of \(C\) with respect to \(A\) and \(B\).
2. [(ii)] Given four points \(A, B, C, D\) on a line \(l\) and three points \(A', B', C'\) on a line \(l'\) in the same plane, find \(D'\) on \(l'\) such that the ranges \((ABCD)\), \((A'B'C'D')\) have the same cross-ratio.
3. [(iii)] Given five collinear points \(A, A', B, B', C\), find \(C'\) such that the pairs \((A, A')\), \((B, B')\), \((C, C')\) are in involution.
4. [(iv)] Given two lines \(l, l'\), which meet at an inaccessible point, and a point \(P\), construct the line joining \(P\) to the point of intersection of \(l, l'\).

A uniform string of weight \(w\) per unit length hangs freely under gravity, with its two ends fastened to fixed supports. Show that the difference in the tension at any two points is \(w\) times the corresponding difference in the heights of the points. \par Prove that the same result is true for a string in contact with a smooth cylinder of any form of section; the string lies in a vertical plane perpendicular to the generators, which are horizontal.

The points \(D, E, F\) lie on the sides \(BC, CA, AB\) respectively of a triangle \(ABC\). Prove that a necessary and sufficient condition for \(D, E, F\) to be collinear is that \[ \frac{BD}{DC} \frac{CE}{EA} \frac{AF}{FB} = -1. \] If \[ \frac{BD}{DC} = -\frac{m}{n}, \quad \frac{CE}{EA} = -\frac{n}{l}, \quad \frac{AF}{FB} = -\frac{l}{m}, \] prove that \[ \frac{EF}{l(m-n)} = \frac{FD}{m(n-l)} = \frac{DE}{n(l-m)}. \]

State (without proof) Descartes' rule of signs connecting the number of positive roots of an algebraic equation with the signs of the coefficients, and deduce a similar rule for the number of negative roots. \par Find the numbers of positive and negative roots of the equations \begin{align*} x^4 - x^3 + x^2 - 1 &= 0, \\ x^8 - x^3 - x^2 + 1 &= 0. \end{align*}

Obtain the tangential equation of a conic in the form \[ Al^2 + Bm^2 + Cn^2 + 2Fmn + 2Gnl + 2Hlm = 0, \] and find equations giving the coordinates of the foci. Explain how to find the condition that the conic shall be a parabola. \par Given an ellipse and two points \(P, Q\) on it, prove that there is a parabola which touches the axes of the ellipse, the normals at \(P\) and \(Q\), and the chord \(PQ\).