Problems

A train of weight \(W\) is travelling with velocity \(v\) when the brakes are applied. The braking force increases at a constant rate from zero to a maximum \(\frac{W}{n}\) which is reached in a time \(T\), after which it remains constant until the train stops. Find the distance \(d\) travelled by the train before it comes to rest, and the time taken to stop the train. \par Sketch the graph of the variation of \(d\) with \(v\).

Two particles of masses \(M\) and \(m\) (\(M>m\)) are placed on the two smooth faces of a light wedge which rests on a smooth horizontal plane. The faces of the wedge are inclined to the horizontal at angles \(\alpha\) and \(\beta\) respectively. Initially the system is at rest. Shew that the smaller particle will move up the plane on which it is placed if \[ \tan\beta < \frac{M\sin\alpha\cos\alpha}{m+M\sin^2\alpha}. \]

Equal particles of mass \(m\) are attached to the ends of a light string \(AB\) which passes through a small smooth fixed ring \(O\), and rest on a smooth horizontal plane. The system is at rest with \(OA\) and \(OB\) taut, and \(OA=OB\). An impulse \(P\) is applied to the particle at \(A\) in a direction making an angle of \(60^\circ\) with the direction \(OA\). Find the initial motion of the particles, and shew that when the particle \(B\) reaches the ring its velocity is \(\frac{\sqrt{22}P}{8m}\).

Two particles \(A\) and \(B\) of masses \(m_1\) and \(m_2\) respectively are connected by a light spring. Prove that if the only force acting on each particle is due to the tension or thrust in the spring, the centre of mass describes a straight line with constant velocity. \par Shew further that the kinetic energy of the system is \[ \frac{1}{2}(m_1+m_2)V^2 + \frac{1}{2}\frac{m_1m_2}{m_1+m_2}v^2, \] where \(V\) is the velocity of the centre of mass, and \(v\) is the velocity of one particle relative to the other. \par The system rests on a smooth horizontal plane, and at time \(t=0\) the spring is in its natural position, the particle \(A\) is at rest, and \(B\) moves with velocity \(u\) in the line of the spring, away from \(A\). If \(m_1=m_2\), and the tension in the spring when its length is increased by unit distance is \(\frac{1}{2}m_1n^2\), shew that at time \(t\) the displacement of \(B\) is \[ \frac{1}{2}u\left(t+\frac{1}{n}\sin nt\right). \]

If \(p\) is a positive integer, shew that the number of distinct ways in which four positive (non-zero) integers may be chosen such that two and only two of them are equal and such that the sum of the four integers is \(12p+1\) is \(p(18p-7)\).

By means of the equation \((x+b)(x+c)-f^2=0\), prove that the equation in \(x\) \[ \begin{vmatrix} x+a & h & g \\ h & x+b & f \\ g & f & x+c \end{vmatrix} = 0 \] has three real roots which are separated by the two roots of the first equation. It may be assumed that \(a,b,c,f,g,h\) are all real and different from zero.

(i) Shew that \[ x - \frac{x^3}{3} + \dots + \frac{x^{4r+1}}{4r+1} > \tan^{-1}x > x - \frac{x^3}{3} + \dots - \frac{x^{4r+3}}{4r+3}, \] where \(r\) is a positive integer and \(0 < \tan^{-1}x < \frac{\pi}{4}\). \par (ii) Shew that, for real quantities, \[ (a_1^2+a_2^2+\dots+a_n^2)(b_1^2+b_2^2+\dots+b_n^2)\dots(k_1^2+k_2^2+\dots+k_n^2) \] is not less than the square of \[ (a_1b_1\dots k_1) + (a_2b_2\dots k_2) + \dots + (a_nb_n\dots k_n). \]

By expanding the function \(x^{n-r}(e^x-1)^r\), prove that for positive integral values of \(s\) less than \(r\) \[ c_0(n-r)^s + c_1(n-r+1)^s + \dots + c_r n^s = 0, \] \(r\) being a positive integer and \(c_0, c_1, \dots c_r\) the coefficients in the expansion of \((1-x)^r\). \par Hence, or otherwise, prove that a series whose \(n\)th term is a polynomial of degree \(r-1\) in \(n\) is a recurring series whose scale of relation is given by \(c_0, c_1, \dots c_r\).

If the circumradius \(R\), and the area \(\Delta\), of a triangle \(ABC\) are regarded as functions of \(b, c, A\), prove that \[ \frac{\partial R}{\partial b}\frac{\partial R}{\partial c} = 4R\sin A \frac{\partial R}{\partial A}, \] \[ \frac{\partial\Delta}{\partial b}\frac{\partial\Delta}{\partial c} + \frac{\partial\Delta}{\partial c}\frac{\partial\Delta}{\partial b} = \frac{R\sin A}{2}. \]

The ordinate of any point on a curve is equal to a cubic polynomial in the abscissa. The curve touches \(Ox\) at the origin and intersects that axis again at the point \((2,0)\). Prove that the tangent to the curve at its point of inflexion cuts \(Ox\) at the point \((\frac{2}{3},0)\). \par Sketch the curve.