Taking as rectangular coordinates the kinetic energy of a particle moving in a straight line and the reciprocal of the resultant force acting on it, and drawing a curve to represent the motion, shew that the distance travelled is represented by the area under this curve. Sketch the curve for the case of a vehicle driven by an engine which is working at constant horse-power, the only resistance being a force proportional to the square of the velocity. Shew that the velocity has a certain limiting value; and calculate the distance travelled while the velocity increases from a half to three quarters of this limiting value, taking the mass of the vehicle as 2000 pounds, the horse-power as 100, and the limiting velocity as 150 feet per second.
Prove, by use of the identity \[ \frac{1}{1+x+x^2} = \frac{1-x}{1-x^3}, \] or otherwise, that \[ 1 - (n-1) + \frac{(n-2)(n-3)}{1.2} - \frac{(n-3)(n-4)(n-5)}{1.2.3} + \dots \] (the series being continued so long as all the factors in the numerator are positive) is equal to \(-1, 0, \text{ or } 1\): and give rules for distinguishing between the different cases.
A light string \(ABCDE\), of length 100 inches, is divided into four equal parts at \(B, C, D\). The ends \(A\) and \(E\) are fixed in the same horizontal at a distance 88 inches apart, and weights \(W, W'\) and \(W\) are hung from \(B, C\) and \(D\). If in equilibrium the depth of \(C\) below \(AE\) is 22 inches, shew that \(W:W' = 11:14\), and that the tensions in \(AB\) and \(BC\) are in the ratio 6:5.
Find the sum of the cubes of the first \(n\) natural numbers. Find the sum to \(2n+1\) terms of the series \(1^3-2^3+3^3-4^3+\dots\).
The equation of a rational algebraic curve of the \(n\)th degree being written in the form \[ x^n f_0\left(\frac{y}{x}\right) + x^{n-1} f_1\left(\frac{y}{x}\right) + x^{n-2} f_2\left(\frac{y}{x}\right) + \dots = 0, \] obtain the asymptotes of the curve; and find the conditions that it may have (i) an inflexional asymptote, (ii) two parallel asymptotes, (iii) a parabolic asymptote. Trace the curve \(x(y^2-4x^2)-x-2y+3=0\), and shew that \((\frac{1}{2},2)\) is a double point.
A well-known safety device for lifts consists of an extension of the lift shaft below ground level; the floor of the lift is made to fit this well closely, so that a pneumatic buffer is provided. A lift weighing 3000 lbs. falls from a height of 30 ft. above ground level into such a safety pit 10 ft. deep, the base of the lift being 8 ft. by 5 ft. Find approximately how far the lift will descend before it is stopped, neglecting air leakage and assuming that the pressure of the air varies inversely as its volume. The resulting equation may be solved graphically: take atmospheric pressure as 15 lbs. per sq. in.
Prove that \[ \begin{vmatrix} 1 & 1 & 1 \\ \sec A & \sec B & \sec C \\ \cosec A & \cosec B & \cosec C \end{vmatrix} = 16 \frac{\sin\frac{B-C}{2} \sin\frac{C-A}{2} \sin\frac{A-B}{2}}{\sin 2A \sin 2B \sin 2C} \{\sin(B+C) + \sin(C+A) + \sin(A+B)\}. \]
A tripod, formed of three equal rods each of weight \(2W\) smoothly hinged together at one end, stands on a smooth horizontal plane, and a weight \(W\) is suspended from the hinge. Equilibrium is maintained by light strings joining the feet \(A, B, C\) of the rods. Shew that the pressure on the plane at \(A\) is \(W(3-2\cot B \cot C)\), and that the tensions of the strings are given by \(T_1 \cos A = T_2 \cos B = T_3 \cos C = 2W \tan\theta \cot A \cot B \cot C\), where \(\theta\) is the inclination of each rod to the vertical.
Find correct to two decimal places the real root of the equation \[ x^3+x^2+x-100=0. \]
Write an essay on the determination of the state of stress in a plane frame built up by light rigid bars, which are smoothly jointed. Consider (i) the relation between the number of joints and the number of bars in a frame, which is just stiff,