A train of weight \(M\) lb. moving at \(v\) feet per second on the level is pulled with a force of \(P\) lb. against a resistance of \(R\) lb. Show that in accelerating from \(v_0\) to \(v_1\) feet per second, the distance in feet described by the train is \(\int_{v_0}^{v_1} \frac{M}{g}\frac{v\,dv}{P-R}\). If the resistance \(R=a+bv^2\), find an expression for the distance described when the power \(P\) is shut off and the velocity decreases from \(v_0\) to \(v_1\).
A car takes a banked corner of a racing track at a speed \(V\), the lateral gradient \(\alpha\) being designed to reduce the tendency to side-slip to zero for a lower speed \(U\). Show that the coefficient of friction necessary to prevent side-slip for the greater speed \(V\) must be at least \[ \frac{(V^2-U^2)\sin\alpha\cos\alpha}{V^2\sin^2\alpha+U^2\cos^2\alpha}. \]
Prove that \(v\frac{dv}{ds}\) and \(v^2/\rho\) are the tangential and normal components of the acceleration of a particle moving with velocity \(v\) in a plane curve. A particle moves in the curve \(y=a\log\sec\frac{x}{a}\) in such a way that the tangent to the curve rotates uniformly; prove that the resultant acceleration of the particle varies as the square of the radius of curvature. Find the period of oscillation of a particle which moves in a straight line under the action of a force directed to a fixed point in the line and proportional to the distance from that point. Two equal particles connected by an elastic string which is at its natural length and straight, lie on a smooth table, the string being such that the weight of either particle would produce in it an extension \(a\). Prove that if one particle is projected with velocity \(u\) directly away from the other, each will have travelled a distance \(\pi u \sqrt{\frac{a}{8g}}\) when the string first returns to its natural length.
The tangents at \(B, C\) to the circumcircle of a triangle \(ABC\) meet in \(L\); \(AL\) cuts the circle in \(P\); and \(Q\) is the mid-point of \(AP\). Prove that \(AQ\) bisects the angle \(BQC\), and that the triangles \(QAB, QCA\) are similar.
Prove that, if \(SY, HZ\) are the perpendiculars from the foci \(S, H\) on the tangent to an ellipse at any point \(P\), then \(Y\) and \(Z\) lie on the circle whose diameter is the major axis \(AA'\). Prove also that, if the line drawn from the centre \(C\) perpendicular to the tangent at \(P\) cuts \(SP, HP\) in \(Q\) and \(R\), then (1) \(SQ=HR=CA\); and (2) \(YQZR\) is a rhombus having its sides equal to \(SC\).
Solve the equations \[ x^2+y^2-3x+1=0, \quad 3y^2-xy+2x-2y-3=0. \]
Prove that \[ \frac{3^3}{1.2} + \frac{5^3}{1.2.3} + \frac{7^3}{1.2.3.4} + \dots = 21e. \]
Prove that, if \(A+B+C+D=\pi\), \[ \cos 2A+\cos 2B - \cos 2C - \cos 2D = 4(\cos A\cos B\sin C\sin D - \sin A\sin B\cos C\cos D). \]
\(O\) is the circumcentre of a triangle \(ABC\) and \(AO, BO, CO\) cut the sides \(BC, CA, AB\) in \(X, Y, Z\). Prove that the ratio of the area of the triangle \(XYZ\) to the area of the triangle \(ABC\) \[ = 2OX.OY.OZ : R^3. \]
Through a point \(P(\alpha,\beta)\) a pair of lines are drawn parallel to the lines \[ ax^2+2hxy+by^2=0 \] which cut the axis \(Ox\) in \(X, X'\) and the axis \(Oy\) in \(Y, Y'\). Find the equation of the line joining the mid-point of \(XX'\) to the mid-point of \(YY'\); and prove that, if this line is perpendicular to the line joining \(P\) to the origin, the locus of \(P\) is a pair of perpendicular lines passing through the origin.