A straight river of width \(d\) flows with uniform speed \(u\). A man, who can swim with constant speed \(v\) (\(v > u\)) and run with constant speed \(w\), starts from a point \(P\) on one bank of the river. He wishes to reach the point \(Q\) on the other bank directly opposite to \(P\). Show that, if he swims across in a straight line, he will take a time \(T_0 = d(v^2-u^2)^{-\frac{1}{2}}\). Find the total time \(T(t)\) that he would take if he first runs upstream for a time \(t > 0\) and then swims to \(Q\) in a straight line. Hence show, by considering \(dT/dt\) or otherwise, that if \(uw > v^2-u^2\) then \(T(t) < T_0\) for sufficiently small values of \(t\).
A sequence of functions \(P_n(x)\), \(n = 0, 1, 2, \ldots\), is defined by setting \begin{align*} P_0(x) &= 1, \quad P_1(x) = x,\\ nP_n(x) &= (2n-1)xP_{n-1}(x) - (n-1)P_{n-2}(x) \end{align*} and requiring \begin{equation*} P_n(x) = \sum_{r=0}^{n} A(n, r)x^r. \end{equation*} if \(n \geq 2\). Show that \(P_n(x)\) is a polynomial of degree \(n\), say Construct a flow diagram for the evaluation of the coefficients in \(P_N(x)\) for a given value of \(N \geq 2\).
The atmosphere at a height \(z\) above ground level is in equilibrium with density \(\rho(z)\). Neglecting the curvature of the earth, show that the pressure \(p(z)\) is given by \begin{equation*} \frac{dp}{dz} = -\rho g, \end{equation*} where \(g(z)\) is the acceleration due to gravity at a height \(z\). If the earth is now assumed to be spherical, it can be shown that the above still holds and that \(g\) is inversely proportional to the square of the distance from the centre of the earth. Assuming also that \(p\), \(\rho\), \(T\) are connected by the relations \begin{equation*} p = k\rho^\gamma, \quad p = R\rho T, \end{equation*} where \(T(z)\) is the temperature of the atmosphere at a height \(z\) and where \(k\), \(R\) are constants with \(\gamma > 1\), show that \begin{equation*} T = T_0\left(1 - \frac{(\gamma-1)a\rho_0g_0z}{(\gamma R\rho_0a+z)}\right), \end{equation*} where \(a\) is the radius of the earth and \(p_0\), \(\rho_0\), \(T_0\), \(g_0\) denote the values of \(p\), \(\rho\), \(T\), \(g\) at \(z = 0\).
A point \(P\) with position vector \(\mathbf{p}(t)\) at time \(t\) moves in a plane in such a way that \begin{equation*} \mathbf{p}\cdot\dot{\mathbf{p}} = 0 \quad \text{and} \quad \ddot{\mathbf{p}} = -\lambda(t)\mathbf{p}, \quad \text{for all} \, t, \end{equation*} where dots denote differentiation with respect to \(t\). If \(P\) is initially at unit distance from the origin, describe its subsequent motion and show that \(\lambda(t)\) is constant. Points \(Q\) and \(R\), with position vectors \(\mathbf{q}(t)\) and \(\mathbf{r}(t)\), move in the same plane so that \begin{equation*} \dot{\mathbf{q}} = \mathbf{k} \quad \text{and} \quad \mathbf{r} = \mathbf{p} + \mathbf{q}, \end{equation*} where \(\mathbf{k}\) is a constant unit vector. Find the conditions required to make \(\mathbf{r}(t)\) vanish at some \(t\), and describe the possible motions of \(R\).
A firm needs to buy a large number of metal links which must stand a load of 1.20 tons weight. There are two grades on the market, the load under which a link will break being in each case normally distributed with parameters as follows:
A ball is dropped from rest at time \(t = 0\) and falls a distance \(a\) on to a horizontal plane. If the coefficient of restitution between the ball and the plane is \(e\), show that the ball will come to rest at time \begin{equation*} \left(\frac{1+e}{1-e}\right)\sqrt{\frac{2a}{g}}. \end{equation*} Suppose now that the experiment is repeated with the plane attached to an apparatus that causes it to oscillate vertically about its previous position in such a way that its displacement, measured vertically downwards, at time \(t\) is \(b\sin \omega t\). Show that for certain values of \(b\) and \(\omega\), a steady motion is maintained, the ball bouncing back to its original position once each period.