A rocket continuously ejects matter backwards with velocity \(c\) relative to itself. Show that if gravity is neglected the velocity \(v\) and total mass \(m\) of the rocket are related by the equation \[ m\frac{dv}{dt} + c\frac{dm}{dt} = 0. \] Deduce that whatever the rate of burning of the rocket, \(v\) and \(m\) are related by the formula \[ v=c\log(M/m), \] where \(M\) is a constant. Assuming that \(m\) decreases at a constant rate \(k\), show that the distance the rocket travels from rest before the mass has fallen from the initial value \(m_0\) to \(m_1\) is \[ c(m_1/k)\{m_0/m_1 - 1 - \log(m_0/m_1)\}. \]
Six shoes are taken at random from a set of a dozen different pairs. What is the probability that they contain at least one pair?
What do you mean by (a) a finite limit and (b) an infinite limit? Evaluate the following limits:
(i) Show that \[ |(x+y)e^{-2(x^2-xy+y^2)}| \le e^{-1/2} \] for all real \(x, y\). When is the sign of equality required? (ii) If \(0< x< pr, 0< y< pr, 0< xy< p^2\), show that \[ x+y < \left(r+\frac{1}{r}\right)p. \]
Establish the formula for the centre of curvature in Cartesian co-ordinates for the curve \(x=x(t), y=y(t)\), where \(t\) is a parameter. Show that the centres of curvature of \[ x=t+\sin t, \quad y=1+\cos t \] lie on \[ x=t-\sin t, \quad y=-1-\cos t. \]
Let \(m\) be a positive integer and \(y \ne \pm 1\). Put \[ (m,0)=1; \quad (m,j) = \frac{(1-y^{2m})(1-y^{2m-2})\dots(1-y^{2m+2-2j})}{(1-y^2)(1-y^4)\dots(1-y^{2j})} \quad (0< j\le m). \] Show that, for \(0 \le j < m\), \[ (m+1, j+1)=(m,j+1)+y^{2m-2j}(m,j), \] and hence, or otherwise, that \[ (m,0)+(m,1)y+(m,2)y^2+\dots+(m,j)y^j+\dots+(m,m)y^m = (1+y)(1+y^3)\dots(1+y^{2m-1}). \] What corresponds to this identity for \(y = \pm 1\)?
The ends of a uniform heavy chain of length \(l\) are attached to light rings threaded on two smooth rods \(OA, OB\). which are fixed in the same vertical plane and make equal angles \(\alpha\) (\(<\frac{1}{2}\pi\)) with the downward vertical through \(O\). Show that the chain can rest in equilibrium under gravity with the rings at the same level and at a distance \(kl\) apart, where \(k\) is such that \[ \sinh(k \tan\alpha) = \tan\alpha. \] Show that, given \(\alpha\), \(k\) is determined uniquely by this equation.
A parallelogram \(ABCD\) of freely jointed rods is in equilibrium on a smooth horizontal table. If \(T_1, T_2\) are the tensions in two strings, of which the first joins a point \(P\) of \(AB\) to a point \(Q\) of \(CD\), while the second joins a point \(R\) of \(BC\) to a point \(S\) of \(DA\), prove that \[ \frac{(AP-DQ)}{AB} \frac{T_1}{PQ} = \frac{(BR-AS)}{DA} \frac{T_2}{RS}, \] and explain the significance of this equation if \((AP-DQ)/(BR-AS)\) is zero or negative.
A particle of unit mass is projected vertically upwards in a medium whose resistance is \(k\) times the square of the velocity of the particle. If the initial velocity is \(V\), prove that the velocity \(u\) after rising through a distance \(s\) is given by \[ u^2 = V^2 e^{-2ks} + \frac{g}{k}(e^{-2ks}-1). \] Deduce an expression for the maximum height of the particle above its point of projection.
The two ends \(A\) and \(B\) of a uniform rod of length \(2a\) and mass \(m\) are attached by light rings to a smooth vertical wire and a smooth horizontal wire respectively. The wires are fixed in space so that the shortest distance between them is equal to \(a\). The rod is released from rest with the end \(A\) at a vertical height \(a\) above the level of the horizontal wire. Find the speed of \(A\) when the rod \(AB\) becomes horizontal.