Define the terms work, energy, and power. A motor-car can travel with speed \(U\) up a slope of 1 in \(a\) and with speed \(V\) down a slope of 1 in \(b\), the power of the engine remaining constant. If the resistance is proportional to the square of the speed, find the maximum speed attainable on a level road. Find also equations for the speed attainable up a slope of 1 in \(c\), and down such a slope. [A slope of 1 in \(a\) is here defined as a slope where the car rises 1 ft. in travelling \(a\) ft. along the road.]
Show that the path of a projectile under gravity is a parabola, and explain the assumptions involved in establishing this. A gun with its barrel inclined at angle \(\alpha\) to the horizontal is wheeled on to a place inclined at angle \(\beta\) to the horizontal, the axles of the gun-carriage being parallel to the lines of greatest slope. Show that the time of flight before a shell fired with velocity \(V\) strikes the plane is \(2V \sin \alpha \sec \beta/g\), and that the direction of the point of impact makes an angle \(\tan^{-1}(\tan \alpha \tan \beta)\) with the horizontal line along the plane through the gun. Find also the ratio of the range to that on a horizontal plane.
Two spheres of masses \(m_1\) and \(m_2\) move with their centres travelling on the same line with velocities \(u_1\) and \(u_2\) and collide. If the coefficient of restitution is \(e\), find the changes of speed produced by the impact. If the initial velocities of the spheres are interchanged and reversed in direction, show that the impulse between the spheres is the same as in the first case. A spherical particle is released at a height \(h\) above a horizontal table with which the coefficient of restitution is \(e\). Show that its average speed, with regard to time, while it is being reduced to rest is \[(\tfrac{1}{2}gh)^{\frac{1}{2}}{(1 + e^2)(1 + e)^{-2}}.\] Explain why this formula does not apply in the case when \(e = 1\).
A particle \(P\) moves in an ellipse under the action of a force directed to a focus \(S\). Show that the force must vary inversely as the square of the distance \(SP\) from the focus. If \(H\) denotes the other focus, show that the speed of the particle varies as \(|HP|/SP|^{\frac{1}{2}}\). If \(A\) denotes the end of the major axis nearer to \(S\), and \(\theta\) denotes the angle \(ASP\), find an expression in terms of \(\theta\) for the time taken to travel from \(A\) to \(P\) as a proportion of the period \(T\) required to describe the whole ellipse. [It may be found helpful to use the eccentric angle of \(P\) on the ellipse.]
Let $$f(x) = 1 + \frac{x}{a} + \frac{x^2}{a(a+1)} + \ldots + \frac{x^n}{a(a+1)\ldots(a+n-1)} + \ldots,$$ where \(|x| < 1\) and \(a\) is a constant satisfying \(0 < a < 1\). Show that \(f(x)\) can be expressed in a form using only the elementary functions and a finite number of operations of addition, subtraction, multiplication, division, integration and differentiation.
Let \(N_+\), \(N_-\) be the number of positive integers of the form \(3k + 1\), \(3k - 1\), respectively, with integral \(k\), which divide a given positive integer \(n\), both 1 and \(n\) being counted. Show that $$N_+ \geq N_-.$$
On a level plain are to be seen three church steeples of different heights. Three men walk on the plain so that each man always sees two of the steeples at equal angles of elevation, no two of the men looking at the same two steeples. Prove that each man walks in a circle and that the centres of the three circles lie in a straight line.
By the 'first octant' of 3-dimensional space with a given co-ordinate system we mean the set of points \((x, y, z)\) with $$x \geq 0, \quad y \geq 0, \quad z \geq 0.$$ A line \(\lambda\) passes through the origin and contains no other point of the first octant. Show that there is a plane \(\pi\) which passes through \(\lambda\) and contains no point of the first octant except the origin.
The tangents at two points \(X\), \(Z\) of a non-singular conic \(S\) meet in \(Y\), and another non-singular conic \(S'\) touches \(XZ\) at \(X\) and touches \(YZ\) at a point distinct from \(Y\). From a general point \(P\) of \(S\) tangents are drawn to \(S'\), meeting \(S\) again in \(Q\), \(R\). Prove that \(QR\) touches \(S'\).
A set of \(m + 1\) white mice is taken at random, where \(m\) and \(n\) are positive integers. Show that at least one of the following two situations must occur: either there is a set of \(n + 1\) white mice or \((1 \leq j \leq n + 1)\) such that \(w_j\) is a parent of \(w_{j+1}\) \((1 \leq j \leq n)\) or there is a set of \(m + 1\) white mice no one of which is a parent of any other.