A wedge of mass \(m\) with its two faces inclined at an angle \(\pi/3\) is at rest on a horizontal plane, the face in contact with the plane being perfectly smooth. It is hit by a uniform right circular cylinder of mass \(m\) and radius \(a\) which is initially rolling along the horizontal plane with the velocity \(a\Omega\), its axis being parallel to the line of intersection of the two faces of the wedge. Assuming that there is no rebound or slipping on impact, show that immediately after the impact, the velocity of the wedge is \(4a\Omega/11\). The cylinder then rolls up the inclined face and back to the horizontal plane; show that the velocity of the wedge when the cylinder reaches the horizontal plane again is \(7a\Omega/11\), and find the final velocity of the cylinder on the horizontal plane, assuming that there is no rebound or slipping (at the point of contact with the plane) on the final impact.
The mid-points of the sides \(AB, CD\) of a parallelogram \(ABCD\) are \(X, Y\). \(P\) is a point on the diagonal \(AC\) and \(PX, PY\) meet \(BC, AD\) respectively in \(U, V\). Prove that \(UV\) is parallel to \(AB\). If \(UV\) cuts \(AC\) in \(W\), prove that \(PW\) is the harmonic mean of \(PA\) and \(PC\), due regard being paid to sign.
Show that by suitable choice of homogeneous coordinates a non-singular conic \(S\) can be expressed in the parametric form \[ x:y:z=t^2:t:1. \] The lines joining the point \(Y(0,1,0)\) to four points \(P_1, P_2, P_3, P_4\) of \(S\) meet \(S\) again in \(Q_1, Q_2, Q_3, Q_4\). Prove that the conics \(YP_1P_2P_3P_4\) and \(YQ_1Q_2Q_3Q_4\) touch at \(Y\), and find, in terms of the parameters of \(P_1, P_2, P_3, P_4\) on \(S\), a necessary and sufficient condition that these conics should also touch at a point other than \(Y\).
Define the greatest common factor of two integers \(m, n\) and describe a method of determining it. The integers \(u_0, u_1, \dots, u_n, \dots\) are defined by \[ u_0=0, \quad u_1=1, \quad u_{r+1}=u_r+u_{r-1} \quad (r \ge 1). \] Prove that, if \(r, s\) are positive integers \[ u_{r+s} = u_r u_{s+1} + u_s u_{r-1} \] and that, if \(s < r\), \[ u_{r-s} = (-1)^s (u_r u_{s+1} - u_s u_{r+1}). \] Deduce that the greatest common factor of \(u_{207}\) and \(u_{345}\) is \(u_{69}\).
Prove that, if \(i^2=-1\) and \(n\) is a positive integer, \[ \left(\frac{1+i\tan\theta}{1-i\tan\theta}\right)^n = \frac{1+i\tan n\theta}{1-i\tan n\theta}. \] Hence, or otherwise, express \(\tan n\theta\) in the form \(\dfrac{f(\tan\theta)}{g(\tan\theta)}\), where \(f(x)\) and \(g(x)\) are polynomials in \(x\). Evaluate \[ \text{(i) } \sum_{k=0}^{k=n-1} \cot \frac{(4k+1)\pi}{4n}; \quad \text{(ii) } \sum_{k=0}^{k=n-1} \cot^2 \frac{(4k+1)\pi}{4n}. \]
Defining \(\log_e t = \int_1^t \frac{du}{u}\), prove that, for \(t>0\), \(\log_e t < t-1\). Let \(f(x ) > 0\) for \(0 < x < 1\). By considering first the special case when \[ \int_0^1 f(t)dt = 1, \] prove the inequality \[ \int_0^1 \log_e f(t)dt \le \log_e \left\{\int_0^1 f(t)dt\right\}. \]
Six uniform straight rods, each of length \(l\) and weight \(W\), are freely jointed at their ends so that they form the edges of a regular tetrahedron. This tetrahedral framework is suspended by a string attached to one vertex. Calculate the horizontal thrust or tension in each of the horizontal rods. Assuming also a linear dependence of stress on vertical displacement in the sloping rods, calculate the longitudinal thrust or tension in one of these rods at a distance \(x (< l)\) from the highest vertex.
A rocket is propelled vertically upwards by the backward ejection of matter at a uniform rate and with constant speed \(V\) relative to the rocket. The total mass of propelling matter available is \(m\) and it is completely ejected at a time \(\tau\) after launching, when the mass remaining to the rocket is \(km\). Show that, when the time \(t\) is less than \(\tau\), the velocity of the rocket varies according to the equation
\[ \frac{dv}{dt} = -g + \frac{V}{(k+1)\tau - t}. \]
If \(g\tau(k+1)
A uniform straight rod \(AB\) of length \(2a\) and mass \(M\) has a particle of mass \(m\) attached at the end \(B\). The rod is moving freely on a smooth horizontal table when it strikes an inelastic peg fixed in the table. Given that the impact reduces the rod to rest, and that just before the impact \(A\) was instantaneously at rest whilst \(B\) was moving with velocity \(V\), find the position on the rod of the point of impact.
Two particles \(A\) and \(B\) each of mass \(m\) are attached to the ends of a light inextensible string of length \(l\). Initially \(A\) and \(B\) are on a smooth horizontal table, so that the string is taut and at right angles to the edge of the table when \(B\) is pushed gently over the edge. Find (i) the time for \(A\) to reach the edge of the table; (ii) the impulsive tension in the string when \(A\) also leaves the table; (iii) the time between \(A\) leaving the table and \(A\) and \(B\) being next at the same horizontal level; (iv) the horizontal distance moved by \(B\) during the time found in (iii); (v) the vertical distance fallen by \(A\) during the time found in (iii).