A particle is projected with velocity \(\sqrt{2ga}\) from a point at a height \(h\) above a level plain. Show that the tangent of the angle of elevation for maximum range on the plain is \(a/(a+h)\), and that the maximum range is \(2\sqrt{a(a+h)}\).
A particle describes a distance \(x\) along a straight line in time \(t\), where \(t=ax^2+bx\), and \(a,b\) are positive numbers. Show the retardation is proportional to the cube of the velocity. If the initial velocity is 2000 feet per second, and is reduced to 1975 feet per second in 100 feet, show that the initial resistance is about 15.8 of the weight of the particle.
A horse pulls a wagon of 10 tons from rest against a constant resistance of 50 lb. The pull exerted is at first 200lb., and decreases uniformly with the distance until it falls to 50lb. after a distance of 167 feet has been covered. Show that the resulting velocity of the wagon is very nearly 6 feet per second.
Find the velocities of two elastic spheres after direct impact with given velocities. Two equal spheres \(A,B\) lie in a smooth horizontal circular groove at opposite ends of a diameter. \(A\) is projected along the groove and at the end of time \(t_0\) impinges on \(B\); show that the second impact will occur at a further time \(\frac{2t_0}{e}\), where \(e\) is the elastic coefficient.
Define simple harmonic motion, and find the velocity in terms of the displacement. A particle is attached to one end of an elastic string whose other end is fixed at \(A\) and whose modulus is equal to the weight of the particle. The particle is let go from rest at \(A\). Show that the greatest extension of the string during the motion is \(1+\sqrt{3}\) times its natural length.
A particle of mass \(m\) is attached to a point \(O\) by an inextensible string of length \(l\). Prove that it can describe a horizontal circle about a point vertically below \(O\) with uniform angular velocity \(\omega\), provided \(l\omega^2 < g\). Prove that if the string is elastic, this condition is replaced by \[ \frac{l_0\lambda\omega^2}{g} > \lambda - ml_0\omega^2 > 0, \] where \(\lambda\) is the modulus of elasticity and \(l_0\) the natural length of the string.
Four coplanar lines, taken in sets of 3, form 4 triangles; prove that the circumcircles of these 4 triangles meet in a point (P) and that their orthocentres lie on a line (l). Prove also that, if Q is a variable point on the line l, the perpendicular bisector of PQ is cut by the 4 original lines in 4 points whose mutual distances are in constant ratios.
The normals at the points \(P,Q\) of an ellipse are perpendicular and meet the ellipse again in \(P',Q'\) respectively. Prove that, if \(C\) is the centre of the ellipse, the sectorial areas \(CPP', CQQ'\) are equal.
If \(\alpha, \beta\) are the roots of the quadratic \[ ax^2+2hx+b+\kappa(a'x^2+2h'x+b')=0, \] prove that numbers \(p,q\), independent of \(\kappa\), can be found such that \((p-\alpha)(p-\beta)=q^2\), and that \((p\pm q)\) are the roots of the quadratic \[ (ax+h)(h'x+b')-(hx+b)(a'x+h')=0. \] Taking \(x\) as the abscissa of any point, give a geometrical interpretation of the preceding result.
Prove that \[ \frac{nx^{2n-1}}{x^{2n}-1} = \frac{x}{x^2-1} + \sum_{r=1}^{n-1} \frac{x-\cos r\alpha}{x^2-2x\cos r\alpha+1}, \] where \(\alpha=\pi/n\); and deduce that \[ \sum_{r=1}^{n-1} \frac{\cos r\alpha}{\cos r\alpha - \cos\theta} = \frac{n\cos(n-1)\theta}{\sin\theta\sin n\theta}-\csc^2\theta. \]