Problems

Three equal smooth uniform spheres \(A, B, C\) lie in that order on a smooth horizontal table, with their centres in a straight line. The coefficient of restitution between any pair of the spheres is \(e\). The sphere \(A\) is projected along the line in the direction of \(B\). Prove that there will be three, and only three collisions between the spheres if \(1 > e \ge 3 - 2\sqrt{2}\). If \(e=3-2\sqrt{2}\), show that the final kinetic energy of the system is approximately one-third of the initial kinetic energy.

A small raindrop falling through a cloud acquires moisture by condensation from the cloud. When the mass of the raindrop is \(m\), the rate of increase of mass per unit time is \(km\), where \(k\) is small. The raindrop starts from rest. Prove that when it has fallen a distance \(h\) through the cloud, its velocity is given approximately by \[ v^2 = 2gh(1-\tfrac{2}{3}k\sqrt{2h/g}). \] (The resistance of the cloud to the motion is to be neglected, and the cloud is assumed to be stationary and of infinite mass.)

A uniform spherical ball of radius \(a\) is at rest on a rough horizontal table, and is set in motion by a horizontal blow in a vertical plane through the centre at a distance \(\frac{2}{3}a\) above the table. Show that when the ball ceases to slip its linear velocity is \(5/14\) of its initial linear velocity.

Find a cubic polynomial in \(x\) which takes the values \[ \frac{1}{a-1}, \frac{1}{a}, \frac{1}{a+1}, \frac{1}{a+2} \] when \(x\) takes the values \(-1, 0, 1, 2\). Verify that \(x-a-1\) is one factor of the polynomial.

If \(n\) and \(s\) are given, show that the product of \(n\) positive integers whose sum is \(s\) is not greater than \(q^{n-r}(q+1)^r\), where \(q, r\) are respectively the quotient and remainder when \(s\) is divided by \(n\).

1. Prove that \(x^2+y^2+z^2-yz-zx-xy\) is a factor of \((y-z)^n+(z-x)^n+(x-y)^n\) provided \(n\) is a positive integer which is not an integral multiple of 3.
2. If \(x+y=a+b+c\) and \(x(x-a)(x-b)(x-c)=y(y-a)(y-b)(y-c)\) and \(x \ne y\), prove that \(x^2+y^2=a^2+b^2+c^2\). (Note: OCR says \(x^3+y^3 = a^3+b^3+c^3\). The image is blurry but looks more like squares.) Let's re-examine. It indeed looks like \(x^2+y^2=a^2+b^2+c^2\) on the scan.

A square of side 6 in. is divided into 36 inch squares. Find the number of paths 12 in. long which join a pair of opposite corners of the 6 in. square and which lie along the sides of the small squares.

Prove for positive values of \(x\), that if \(p>q>0\), then \[ q(x^p-1) \ge p(x^q-1). \] Hence, or otherwise, show that for, \(x \ge 1\), and \(p \ge q > 0\),

1. \(\frac{x^p-1}{p} \ge \frac{x^q-1}{q} - \frac{(p-q)}{pq}\log_e x\).
2. \(\frac{qn}{p-n}(x^p-1) \ge \frac{pn}{q-n}(x^q-1) - \frac{pq(p-q)}{(p-n)(q-n)}(x^n-1)\),

Prove that a necessary condition for the radius of curvature to be equal to the perpendicular from the origin on to the tangent for every point of a curve, is that the intrinsic equation is of the form \(s = \frac{1}{2}(a\psi^2+2b\psi+c)\). Show that with this form of intrinsic equation given, an origin can be found to satisfy the former property. Prove that in this case, the centre of curvature lies on a fixed circle of radius \(a\).

If \(y = \tan^{-1}\frac{x\sin\theta}{1+x\cos\theta}\), where \(\theta\) is constant, show that for \(n \ge 2\), \[ (1+2x\cos\theta+x^2)y_n + 2(n-1)(x+\cos\theta)y_{n-1} + (n-1)(n-2)y_{n-2}=0, \] where \(y_n\) denotes \(\frac{d^n y}{dx^n}\). Deduce that \(y_n(0) = (-1)^{n-1}(n-1)!\sin n\theta\).