Two smooth spheres of equal mass whose centres are moving with equal speeds in the same plane, collide in such a way that at the moment of collision the line of centres makes an angle \(90^\circ-\beta\) with the direction bisecting the angle \(\alpha\) between the velocities before impact. Shew that after impact the velocities are inclined at an angle \(\tan^{-1}(\tan\alpha\cos 2\beta)\), the collision being perfectly elastic.
A heavy particle is attached by a light elastic string to a fixed point \(A\) on a rough plane whose inclination to the horizontal is \(\alpha\). Originally the string is unstretched, and lying along a line of greatest slope, \(A\) being the highest point of the string. Describe the motion in general terms, and shew that it will be all in one direction unless the coefficient of friction is less than \(\frac{1}{2}\tan\alpha\).
The propulsive horse-power required to drive a ship of mass 16,500 tons at a steady speed of 30 feet per second is 18,000. Assuming that the resistance is proportional to the square of the speed, and that the engines exert a constant propulsive force on the ship at all speeds, prove that the initial acceleration, when the ship starts from rest, is \(\frac{1}{5}\) feet per sec. per sec.; and that it attains a speed of 20 feet per sec. in \(\frac{5}{2}\log_e 5\) minutes.
If \[ y = ax\cos\left(\frac{n}{x}+b\right), \] prove that \[ x^4 \frac{d^2 y}{dx^2} + n^2 y=0. \] Prove that \(x^{1/x}\) is a maximum when \(x=e\).
If \[ y=a+x\log y, \] prove that when \(x\) is zero \[ \frac{dy}{dx} = \log a \quad \text{and} \quad \frac{d^2 y}{dx^2} = \frac{1}{a}(\log a)^2. \] If \(\lambda\) is a variable parameter, find the envelope of the family of straight lines \[ x(1+\lambda^2)+2\lambda y = a(1-\lambda^2). \]
Establish the equations of a cycloid in the form \begin{align*} x &= a(\theta+\sin\theta) \\ y &= a(1-\cos\theta) \end{align*} A circle rolls a long distance along a straight line. Prove that the path described by a point on its circumference is longer than the path described by its centre in the ratio \(\frac{4}{\pi}\).
A variable point \(X\) is taken on the side \(BC\) of a quadrilateral \(ABCD\); and the line drawn through \(B\) parallel to \(AX\) cuts in \(P\) the line drawn through \(C\) parallel to \(DX\). Prove that the locus of \(P\) is a fixed line parallel to \(AD\).
Three tangents to a parabola whose focus is \(S\) form the triangle \(ABC\). Prove that the tangent to the parabola that is perpendicular to \(SA\) cuts \(BC\) in the same point as the line joining \(S\) to the centre of the circle \(ABC\).
Prove that \(a^2+b^2+c^2-bc-ca-ab\) is a factor of the expression \[ (b-c)^n+(c-a)^n+(a-b)^n \] if \(n\) is a positive integer which is not a multiple of 3.
If \((1+x)^n = c_0+c_1 x+\dots+c_n x^n\), prove that \[ \frac{c_0}{n+1}-\frac{c_1}{n+2}+\frac{c_2}{n+3}-\dots+(-1)^n\frac{c_n}{2n+1} = \frac{(n!)^2}{(2n+1)!}. \]