A uniform cubical block of wood of edge \(a\) and mass \(M\) rests with one of its faces in contact with a smooth horizontal plane. A small bullet of mass \(m\) travelling with high velocity \(U\) impinges normally on one of the vertical faces of the block. It hits the centre of this face at time \(t = 0\) and penetrates the wood, in which it is subject to a retarding force of magnitude \(kv\), where \(v\) is the velocity of the bullet relative to the block. Show that if $$U > \frac{(M + m)kA}{Mm}$$ the bullet will go right through the block, emerging with velocity $$U - \frac{kA}{m}$$ at time $$t = -\frac{Mm}{(M + m)k}\log\left(1 - \frac{(M + m)kA}{MmU}\right).$$ [Gravity may be ignored.]
A ball is projected towards a smooth high wall from a point at a distance \(a\) from the wall, the plane of projection being perpendicular to the wall. The velocity of projection is \(V\), the angle of projection is \(\alpha\), and the coefficient of restitution is \(e\). Show that, provided \(V^2 > ag(1 + e^{-1})\), there are two values of \(\alpha\) for which the ball returns directly to its starting point after bouncing off the wall. If \(\alpha\) has one of these values and \(\phi\) is the angle of inclination to the horizontal at which the ball then rebounds from the wall, show that $$e(1 + e)\tan \phi = (1 - e)\tan \alpha.$$
A shell of mass \(M\) is at rest in space, when it bursts into two fragments, the energy released being \(E\). Show that the relative speed of the fragments after separation cannot be less than \(2\sqrt{(2E/M)}\). Explain how your conclusion is affected if the shell is moving initially with speed \(U\).
A smooth pulley is fixed to the edge of the roof of a building at a height \(h\) from the ground. A light cord of length \(l\) is passed over the pulley and has two buckets attached to its ends, one of which rests on the ground and one of mass \(3m\) hanging. A man on the roof drops a brick of mass \(m\) into the second bucket and it remains in the sand without bouncing. Find the impulsive tension in the cord. Which of the buckets will hit the ground next, and after how long?
A wire in the form of a circle of diameter \(6a\) is fixed in a vertical plane. A bead of mass \(m\) is connected to the highest point by means of an elastic string of natural length \(3a\) which exerts a force \(\lambda (l - 3a)\) when stretched to length \(l\), where \(\lambda = 2mg/a\). The bead is initially sliding down the wire, and when its angular distance \(2\theta\) from the lowest point is \(120^\circ\), so that the string becomes taut, its speed is \(3\sqrt{(ga)}\). Show that it will continue moving down till it reaches the bottom and that its speed will then be \(4\sqrt{(ga)}\). Find also how long it takes to get there.
A battery \(B\) of voltage \(V\) is connected through a switch \(S\) with a circuit containing a capacity \(C\) and two equal resistances \(R\). The capacitor is initially uncharged, the switch is then closed and is opened again when the charge on \(C\) has reached \(\frac{3}{5}CV\). Show that after an equal time has elapsed the charge will have fallen to \(\frac{9}{25}CV\).
A sample of \(n\) coins is drawn at random from a large collection in which a fraction \(r\) of the \(n\) coins are pennies. What is the probability that just \(r\) of the \(n\) coins are pennies? If the probability that a penny is a Queen Elizabeth one is \(q\), what is the probability that there are exactly \(s\) Queen Elizabeth pennies among the \(r\) pennies of this sample? Write down the probability that a sample of \(n\) coins will contain \(s + k\) pennies, only \(k\) of which are Queen Elizabeth ones, and calculate the sum of these probabilities for all possible values of \(k\).
An investigator collects data on the expenditure in a given week of each of 300 households. He rounds off the figures to the nearest pound and takes the average. Assuming that for any one household the error he thus makes is equally likely to have any value between plus and minus 10 shillings, find the standard deviation of the departure of his answer from the true average.
A factory makes components in the form of a rectangle whose length is intended to be twice its breadth. There is, however, a random error with standard deviation 0.1\% in the lengths; similarly, the breadths are distributed independently about a certain value with standard deviation 0.1\%. Find the percentage standard deviations of the perimeters and of the areas of the components produced.
A \emph{plane convex set} is a set of points in a plane such that any point of the line-segment joining any 2 points of the set also belongs to the set. 4 plane convex sets \(S_1, S_2, S_3, S_4\) are such that \(S_c, S_2, S_3\) have a common point \(P_1\); \(S_3, S_4, S_1\) have a common point \(P_2\); \(S_4, S_1, S_2\) have a common point \(P_3\); and \(S_1, S_2, S_3\) have a common point \(P_4\). By considering separately the three cases: