Problems

Consider a group of students who have taken two examination papers. Suppose that 80\% of these students pass on Paper I. Suppose further that any student who passes on Paper I has a 70\% chance of passing on Paper II while those who failed Paper I have only a 20\% chance. What is the probability that a student who passes on Paper II did not pass on Paper I?

Let the random variable \(X\) have the exponential distribution with parameter \(\lambda > 0\), that is \[P\{X \leq x\} = \begin{cases} 1-e^{-\lambda x}, & \text{if}~ x \geq 0,\\ 0, & \text{if}~ x < 0. \end{cases}\] Let \(Y\) be a random variable having the exponential distribution with parameter \(\mu\), and suppose that \(X\) and \(Y\) are independent. Find the distribution of min\((X, Y)\) and the probability that \(Y\) exceeds \(X\).

Explain what is meant by the parallelogram of forces, and what is meant by the resultant of a system of coplanar forces. \(ABCD\) is a quadrilateral whose opposite sides meet in \(X\) and \(Y\). By considering suitable forces acting along the sides of the quadrilateral, show that the bisectors of the angles \(X\), \(Y\), the bisectors of the angles \(B\), \(D\) and the bisectors of the angles \(A\), \(C\) intersect on a straight line, certain restrictions being made as to which pairs of bisectors are taken.

Two small spherical particles of mass \(m\) are joined by inextensible light strings of length \(a\) to a particle of mass \(M\); the strings lie taut in a straight line on opposite sides of \(M\) on a smooth horizontal table. The particle of mass \(M\) is set in motion by an impulse \(I\) perpendicular to the line of the particles. Show that when the two small spheres collide their relative velocity is \(2I/\sqrt{M(M+2m)}\). The spheres are imperfectly elastic, with coefficient of restitution \(e\). Find the angular velocities of the strings when they are next in line. [You may assume that the strings remain taut throughout the motion.]

A planet moves about the sun under the influence of a radial force \(F(r)\), \(r\) being the distance from the sun to the planet. Show that the differential equation of the orbit can be written in the form \[\frac{d^2u}{d\theta^2} + u = \frac{f(u)}{h^2u^2},\] where \(u = r^{-1}\), \(f(u) = F(r)\), \((r, \theta)\) are polar coordinates of the planet in the plane of the orbit, and \(h\) is a constant. According to special relativity, \(F(r)\) takes the form \[F(r) = -\frac{GM}{r^2}\left(\frac{E}{m_0c^2}+\frac{GM}{rc^2}\right),\] where \(M\) is the sun's mass, \(m_0\) the planet's mass, and \(G\), \(E\) and \(c\) are constants. In the approximation \(GM \ll hc\), find the equation of the orbit and describe the motion.

A ground-to-ground missile leaves its launch pad with speed \(V_0\) at a small angle \(\psi_0\) to the horizontal. The mass \(m\) of the missile and the thrust \(T\) with which it is driven may both be assumed constant throughout the flight, and \(T\) may be assumed to be horizontal. Show that for \(\psi_0 \ll mg/T\):

1. the angle of impact is also \(\psi_0\);
2. the speed at the highest point of the trajectory is approximately \[V_0\left(1+\frac{T\psi_0}{mg}\right);\]
3. in this approximation the impact speed \(V_I\) is given by \[V_I = V_0\left(1+\frac{2T\psi_0}{mg}\right).\]

State clearly any other approximations that you use.

Two particles of equal mass collide. Before the impact, their velocities are \(\mathbf{v}_1\) and \(\mathbf{v}_2\) and afterwards they are \(\mathbf{v}_1'\) and \(\mathbf{v}_2'\). Momentum and energy are conserved. Show that

1. the relative velocity of the particles has the same magnitude before and after the impact;
2. if one particle is initially at rest, the directions of motion after impact are perpendicular to one another.

Now consider the collision of an election of mass \(m\) and velocity \(\mathbf{v}_0\) with a stationary atom of mass \(M\). Momentum is conserved, but an amount \(W\) of kinetic energy is lost. Find a quadratic equation satisfied by the magnitude of \(\mathbf{V}\), the velocity of the atom after the collision, and involving the angle between \(\mathbf{V}\) and \(\mathbf{v}_0\), and deduce that the initial kinetic energy of the election must not be less than \(W(1 + (m/M))\)

A perfectly elastic particle bounces off a smooth wall. Let \(\mathbf{n}\) denote the unit vector normal to the wall and directed away from the wall at the point of impact, \(\mathbf{k}_1\) denotes the unit vector in the direction of motion of the particle immediately prior to impact, and \(\mathbf{k}_2\) denotes the unit vector in the direction of motion immediately after impact, show that \begin{equation} \mathbf{k}_2 = \mathbf{k}_1 - 2(\mathbf{n} \cdot \mathbf{k}_1)\mathbf{n}. \end{equation} Such a particle bounces successively off each of three mutually perpendicular smooth planes. If the particle is acted upon by no forces other than those occurring during impact with the planes, show that the particle emerges travelling parallel to (but in the opposite sense from) its initial direction of motion.

Solve the recurrence relation \[u_{n+2}-2\alpha u_{n+1}+(\alpha^2+\lambda^2)u_n = 0, \quad u_0 = 0, \quad u_1 = 1,\] in the following two cases.

1. When \(\lambda = 0\).
2. When \(\lambda > 0\).

As \(\lambda \to 0\), does your solution to (2) tend to your solution to (1)?

Let \(q\) be an integer. If \(q > 1\) show that every positive real number \(x\) has an expansion to the base \(q\), that is \[x = \sum_{r=0}^{N} a_r q^r + \sum_{s=1}^{\infty} b_s q^{-s}\] where \(N\) is finite, and for each \(r\) and \(s\), \(a_r\) and \(b_s\) are integers satisfying \(0 \leq a_r < |q|\) and \(0 \leq b_s < |q|\). Is this result still true if \(q = -2\)?