A single stream of cars, each of width \(a\) and exactly in line, is passing along a straight road of breadth \(b\) with speed \(V\). The distance between the rear of each car and the front of the one behind it is \(c\). Show that, if a pedestrian is to cross the road in safety in a straight line making an angle \(\theta\) with the direction of the traffic, then his speed must be not less than \[\frac{Va}{c\sin\theta+a\cos\theta}.\] Show also that if he crosses the road in a straight line with the least possible uniform speed, he does so in time \[\frac{b}{V}\left(\frac{c}{a}+\frac{a}{c}\right).\]
A uniform beam of weight \(W\) stands with one end on a sheet of ice and the other end resting against the smooth vertical side of a heavy chair of weight \(\lambda W\). Show that the maximum inclination of the beam to the vertical is given by \(\tan^{-1} 2\mu\) or \(\tan^{-1} 2\lambda\mu\) according as the chair or the beam is the heavier, the coefficient of friction between the ice and beam, and the ice and chair, being \(\mu\).
A horizontal conveyor belt moves with a constant velocity \(u\). At time \(t = 0\), a parcel of mass \(m\) is dropped gently onto the belt. If the coefficient of friction between the parcel and the belt is \(\mu\), find
Two particles of masses \(m\) and \(2m\) are suspended over a movable pulley of mass \(m\) by a light string of length \(l\). The movable pulley is itself connected to a particle of mass \(4m\) by a light string of length \(L\) which passes over a fixed pulley. Find the acceleration of the particle of mass \(4m\). [You may neglect the moments of inertia of the pulleys.]
A uniform spherical dust cloud of mass \(M\) expands or contracts in such a way as to remain both uniform and spherical. The gravitational force on a particle of mass \(m\) at a distance \(r\) from the origin is radial and given by \[F = -\frac{4\pi}{3}G\rho mr,\] \(\rho\) being the density of the cloud and \(G\) the gravitational constant. By considering a particle on the surface of the cloud at distance \(R\) from the centre of the cloud, or otherwise, show that \[\frac{1}{2}\dot{R}^2 - \frac{GM}{R} = -\frac{GM}{R_M},\] \(R_M\) being a constant. Verify that for \(R_M > 0\) this equation has a solution of the form \begin{align*} R &= a\sin^2\chi\\ t &= b(\chi-\sin\chi\cos\chi), \end{align*} where \(a\) and \(b\) are constants. Evaluate \(a\) and \(b\) in terms of \(G\), \(M\) and \(R_M\). Show that this solution describes a cloud that expands from infinite density at \(t = 0\) and which collapses back to infinite density at time \[t_\infty = \pi\sqrt{\frac{R_M^3}{2GM}}.\]
A spherical raindrop has mass \(m\), radius \(r\) and downward speed \(v\) as it falls through a cloud of water vapour, which is moving upwards at speed \(U\). The raindrop grows by the condensation of water vapour on its surface, so that the increase of mass per unit time is proportional to the surface area. The raindrop starts from rest with radius \(r_0\) at time \(t = 0\).
A uniform fine chain of length \(l\) is suspended with its lower end just touching a horizontal table. The chain is allowed to fall freely. If the mass of the chain is \(M\), find the force on the table when a length \(x\) has reached it. [You may assume that the part of the chain on the table does not interfere with the subsequent motion.]
Let \(f(x)\) be a polynomial in \(x\). Explain why, if \(z\) is an approximation to a root of \(f(x)\), then \(z-f(z)/f'(z)\) is often a closer approximation. By considering polynomials of the form \(x^r+a\), and suitable real values of \(z_0\), show that the iteration \[z_n = z_{n-1}-f(z_{n-1})/f'(z_{n-1}) \quad (n = 1, 2, \ldots)\] may exhibit any of the following three behaviours.
Show that, for \(r \geq 10\), \[(r-\frac{1}{2})(r+\frac{1}{2}) < r^2 < (r-\frac{39}{80})(r+\frac{41}{80}).\] Deduce that \[\frac{80}{761} \leq \sum_{r=10}^{\infty} \frac{1}{r^2} \leq \frac{2}{19}.\]
Lady Bracknell is holding a dinner party. She has arranged the six diners around a circular table, with Algernon next to Cecily. It is the custom at Bracknell Hall for those dining to change places several times during the meal, in order to vary the conversation. Let \(G\) be the set of those rearrangements of the six diners after which Algernon and Cecily are still sitting next to each other. (Two rearrangements are to be considered the same if one can be obtained from the other merely by rotating the diners around the table.) Show that \(G\) forms a group, under the operation of performing one rearrangement after another. How many elements does \(G\) have? Now suppose that, in the initial arrangement, Cecily is seated on Algernon's right. Let \(H\) be the set of those elements of \(G\) after which Cecily is still on Algernon's right. Show that whenever \(g\) is an element of \(G\) and \(h\) an element of \(H\), \(ghg^{-1}\) is an element of \(H\).