A particle moves in the \(x\)-\(y\) plane with the following equations of motion: \begin{align} \ddot{x} = y\dot{d}, \quad \ddot{y} = c - x\dot{d} \end{align} where \(c\) and \(d\) are constant. Show that the quantity \(\frac{1}{2}(\dot{x}^2 + \dot{y}^2) - cy\) is constant. At \(t = 0\) the initial conditions are \(x = 0\), \(y = 0\), \(\dot{x} = Q\) and \(\dot{y} = 0\). Show that the motion is along the \(x\)-axis if and only if \(Q\) has a certain value, which is to be determined.
A small body of mass \(M\) is moving with velocity \(v\) along the axis of a long, smooth, fixed, circular cylinder of radius \(L\). An internal explosion splits the body into two spherical fragments, with masses \(qM\) and \((1-q)M\), where \(q \leq \frac{1}{2}\). After bouncing elastically off the cylinder (one bounce each) the fragments collide and coalesce. The collision occurs a time \(5L/v\) after the explosion and at a point \(\frac{3}{4}L\) from the axis. Show that \(q = \frac{3}{8}\). Find the energy imparted to the fragments by the explosion, and find the velocity after coalescence. The effect of gravity may be neglected.
The banks of a straight river are given by \(x = 0\) and \(x = a\) in a horizontal rectangular coordinate system \((x, y)\). The water flows in the positive \(y\)-direction with a speed \(3ux(a-x)/a^2\) which depends on the distance \(x\) from the bank. An otter which swims at a steady speed \(u\) starts from the coordinate origin and swims at a constant angle \(\theta\) to the current. Evaluate \(dy/dx\) for its motion and hence find \(y\) as a function of \(x\). If it arrives at the far bank at the point \((a, 0)\) directly opposite its starting point, show that \(\theta = \frac{2\pi}{3}\). For this case find also the values of \(x\) for which \(|y|\) is maximum.
A particle of mass \(m\) is projected vertically upwards in a medium which resists the motion with a force \(mk v^2\), where \(v\) is the speed of the particle. Show that it reaches its greatest height in a time less than \(\pi/2\sqrt{(kg)}\), where \(g\) is the acceleration of gravity. If its speed of projection is \(u\), find its speed when it returns to ground level.
The females of a particular species of beetle live for at most three years and sexually mature in their second and third years. One eighth of the first-year females survive to their second year, and one half of the second year females survive to a third year. The beetles all mate once a year at the same time. In her second year female who mates with a fertile male produces, on average, 6 female offspring and in her third year she produces 8. Let the populations of first, second- and third-year females in a given year (say year \(n\)) be \(X_n\), \(Y_n\), \(Z_n\) respectively. If all the males are fertile, show that \begin{align} X_{n+1} = 6Y_n + 8Z_n \end{align} and write down the corresponding equations for \(Y_{n+1}\) and \(Z_{n+1}\). Show that these equations have a solution \(X_n = A\lambda^n\), where \(A\) and \(\lambda\) are constants, and \(\lambda\) is a real root of the equation \begin{align} \lambda^3 - \frac{3}{4}\lambda - \frac{1}{2} = 0 \end{align} Deduce that the population can grow indefinitely. Show that the female population can remain constant if one-fifth of the sexually active males are infertile.
A community is made up of \(R\) independent, continuously-varying populations, of which the \(r\)th has population \(N_r\) and constant growth-rate \(k_r\) (i.e. \(dN_r/dt = k_r N_r\)). If \(k\) is the growth-rate of the total population of the community, \(N\), show that \begin{align} \text{(a)} \sum_{r=1}^{R} k(k_r - k)N_r = 0 \\ \text{(b)} \frac{dk}{dt} = \frac{1}{N}\sum_{r=1}^{R} k_r^2 N_r - k^2 \\ \text{(c)} \frac{dk}{dt} \geq 0 \end{align}