Problems

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 \(q = -2\)?

Let \(G\) be the set of all \(n \times n\) matrices such that each row and each column has one 1 and \((n-1)\) zeros. Assuming that matrix multiplication is associative, prove that \(G\) forms a group under multiplication and that it has \(n!\) elements.

Let \(N\) be the set of positive integers and \(f\) a function from \(N\) to \(N\). Define, for \(k \in N\) and \(n \geq 0\), \(f^n(k) = k\), \(f^{n+1}(k) = f(f^n(k))\). If \(k\) and \(l\) are such that there are \(n \geq 0\) and \(m \geq 0\) with \(f^n(k) = f^m(l)\), let \(d(k, l)\) be the least possible value of \(n + m\) for such a pair; otherwise set \(d(k, l) = +\infty\).

1. Show that, with the usual conventions regarding \(+\infty\), \[d(k, l) \leq d(k, j) + d(j, l)\] for all \(k, l, j \in N\).
2. Show that if \(d(k, l) < +\infty\), \[d(k, l) - 1 \leq d(f(k), l) \leq d(k, l) + 1.\] Let \(K_k\) be the set of all \(l\) with \(d(k, l) < +\infty\).
3. Show that if \(j \in K_k\) then \(K_j = K_k\).
4. Show that if \(f(k) = k\) then \(K_k\) is the union of two disjoint sets \(A\) and \(B\) such that if \(l \in A\) then \(f(l) \in B\), and if \(l \in B\) then \(f(l) = k\) or \(f(l) \in A\).

Distinct points \(A\), \(B\) are on the same side of a plane \(\pi\). Find a point \(P\) in \(\pi\) such that the sum of the distances \(PA\), \(PB\) is a minimum, and prove that \(P\) has this property.

Find the coordinates of the mirror image of the point \((h, k)\) in the line \[lx + my + n = 0.\] Show that the rectangular hyperbola \[xy = c^2\] touches the rectangular hyperbola \[xy - 2c(x + y) + 3c^2 = 0,\] and that each is the mirror image of the other in the common tangent.