In a cannery, peas of mass \(M\) come out of a pipe uniformly at a velocity \(V\) with a separation \(d\). A fly of mass \(m\) sitting on the end of the pipe hops with negligible speed on to a passing pea. How much energy must the fly's legs absorb in order to hang on to the pea? How far does the fly travel before the pea behind catches up? Just before the peas collide, the fly lets go of the first pea with no change of velocity. The peas then collide without loss of energy, after which the fly catches hold of the second pea. How fast is the first pea travelling after the collision? How far does the fly travel on the second pea before being caught up by the third pea?
A light inextensible string of length \(aL\) is attached at one end \(C\) to a smooth vertical wall and at the other end \(B\) to a uniform rigid straight rod \(AB\) of mass \(M\) and length \(L\). The end \(A\) rests against the wall; \(A\), \(B\) and \(C\) are not collinear, and the plane \(AB\) is vertical. Determine the inclination of the rod to the vertical and the limits on \(a\) between which equilibrium is possible. Show also that the tension in the string is \[\frac{3MgaL}{|2(a^2-1)|}.\]
Solve the vector equation \[\lambda \mathbf{x} + (\mathbf{x} \cdot \mathbf{a}) \mathbf{b} = \mathbf{c},\] where \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) are given non-zero vectors and \(\lambda\) is neither 0 nor \(-\mathbf{a} \cdot \mathbf{b}\). Derive solutions for the special cases \(\lambda = 0\) and \(\lambda = -\mathbf{a} \cdot \mathbf{b} (\neq 0)\), specifying any conditions needed on \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\).
The atmosphere at a height \(z\) above ground level is in equilibrium and has density \(\rho(z)\). By considering the force balance on a thin layer of the atmosphere and neglecting the curvature of the earth, show that the pressure \(p(z)\) is given by \[\frac{dp}{dz} = -\rho g,\] where \(g\) is the acceleration due to gravity (assumed constant). Hence derive an expression for the pressure in an isothermal atmosphere (in which \(p = k\rho\), where \(k\) is a constant) in terms of the pressure \(p_0\) at the surface of the earth. A large spherical balloon of radius \(a\) and total mass \(m\) floats with its centre at a height \(h\) above the surface of the earth. Show that \(h\) is given by \[e^{\alpha h/k} = \frac{2\pi p_0 k^2}{mg^3}\{e^{\alpha(a-1)}+e^{-\alpha(a+1)}\},\] where \(\alpha = ga/k\).
The real 6-dimensional vector space V consists of all homogeneous quadratics \begin{align*} p(x, y, z) \equiv ax^2 + by^2 + cz^2 + 2dyz + 2ezx + 2fxy \end{align*} in \(x, y\) and \(z\), under the usual definitions of addition and multiplication by scalars. Find the dimension of, and write down a basis for,
Positive rational 'weights' \(m_1, \ldots, m_n\) are attached to positive numbers \(a_1, \ldots, a_n\). Use the inequality connecting the arithmetic and geometric means to prove that \begin{align*} \frac{m_1a_1 + \ldots + m_na_n}{m_1 + \ldots + m_n} \geq (a_1^{m_1} a_2^{m_2} \ldots a_n^{m_n})^{1/(m_1 + \ldots + m_n)}. \end{align*} By attaching suitable weights to 1 and \(1 + x/n\), prove that, if \(x\) is positive, \begin{align*} \left(1 + \frac{x}{n+1}\right)^{n+1} \geq \left(1 + \frac{x}{n}\right)^n. \end{align*}
The real polynomial \(f(x)\) has degree 5. Prove that \begin{align*} \int_{-y}^{+y} f(x) dx = \frac{1}{2}y\{f(\lambda y) + f(\mu y) + f(-\lambda y) + f(-\mu y)\} \end{align*} for positive constants \(\lambda\) and \(\mu\) (independent of \(y\) and \(f\)) whose squares are the roots of a certain quadratic equation to be determined.
From the circumcentre \(S\) of a triangle \(ABC\), perpendiculars \(SD\), \(SE\) and \(SF\) are drawn to the sides \(BC\), \(CA\) and \(AB\) respectively, and produced to \(A'\), \(B'\) and \(C'\) so that \(D\), \(E\) and \(F\) are the mid-points of \(SA'\), \(SB'\) and \(SC'\). Prove that the triangles \(ABC\) and \(A'B'C'\) are congruent, that \(AA'\), \(BB'\) and \(CC'\) all have a common mid-point \(M\), and that a rotation about \(M\) moves one triangle to the other.
Suppose \(a > b > 0\). Show that the circle of curvature of the ellipse \begin{align*} x^2/a^2 + y^2/b^2 = 1 \end{align*} at the point \((0, -b)\) is \begin{align*} b^2x^2 + (by + b^2 - a^2)^2 = a^4. \end{align*} The tangent to the ellipse at \((a \cos \theta, b \sin \theta)\) meets the circle at \(P\) and \(Q\), and the lines from \((0, -b)\) to \(P\) and \(Q\) meet the \(x\)-axis at \(X_1\) and \(X_2\). Show that the distance from \(X_1\) to \(X_2\) is equal to the distance between the foci of the ellipse.
A sequence of numbers \(x_0, x_1, \ldots\) is defined by \begin{align*} x_0 &= 0,\\ x_{n+1} &= x_n + \frac{1}{2k}(x^{2k} - x_n^{2k}), \end{align*} where \(-1 \leq x \leq 1\) and \(k\) is a positive integer. Show that \begin{align*} x_0 \leq x_1 \leq \ldots \leq x_n \leq x_{n+1} \leq \ldots \leq |x| \end{align*} and find the limit of the sequence \(x_0, x_1, x_2, \ldots\).