A rocket in rectilinear motion is propelled by ejecting all the products of combustion of the fuel from the tail at a constant rate and at a constant velocity relative to the rocket. Show that, for a given initial total mass \(M\), the final kinetic energy of the rocket is greatest when the initial mass of fuel is \((1-e^{-2})M\).
The ends \(A\), \(B\) of a light rod \(AB\) are joined by light inextensible strings \(AO\), \(BO\) to a fixed point \(O\), and \(AO\) and \(BO\) are equal in length and perpendicular to each other. If weights \(W_1\) and \(W_2\) are now suspended from \(A\) and \(B\), find the angle to the horizontal that the rod will take up in equilibrium.
A uniform chain of length \(b\) and weight \(w\) per unit length has one end free to slide on a smooth vertical wire and passes over a smooth peg at distance \(a\) from the wire, the whole system being in a vertical plane. To the other end of the chain is attached a weight \(wm\). Show that for equilibrium to be possible \(n + b/a\) must be not less than \(e\).
A loop of light inextensible string \(OABCO\) passes in a vertical plane over a horizontal circular cylinder, and a weight \(W\) is attached at \(O\). The portions \(OA\), \(OC\) are straight and perpendicular to each other and tangential to the cylinder at \(A\) and \(C\), and the weight hangs in equilibrium. A gradually increasing couple is applied to the cylinder and slipping is found to occur when it attains the value \((5 \sqrt{2}/13) W \cdot OA\). Find the coefficient of friction between the string and cylinder.
The angle of elevation of a point \(P\) from an origin \(O\) is \(\theta\), and a particle is projected under gravity from \(O\) with given speed \(V\) to pass through \(P\). Show that in general there are two possible trajectories, and that if \(z_1\) and \(z_2\) are the two angles of projection \(z_1 + z_2 = 90^\circ + \theta\). Prove also that if \(T_1\) and \(T_2\) are the corresponding times required to reach \(P\), then \(gT_1T_2\) depends only on the distance \(OP\).
Solution: Suppose we are projected with speed \(V\) at angle \(\alpha\), then: \(u_x = V \cos \alpha, u_y = V\sin \alpha - g t\) \(x= V \cos \alpha t, y= V \sin \alpha t - \frac12 g t^2 \Rightarrow t = \frac{x}{V \cos \alpha}\) So: \(y = x\tan \alpha - \frac12 \frac{g}{V^2} \sec^2 \alpha \cdot x^2\) Suppose are at the point \(P\) at \((k, k \tan \theta)\), then we must have: \begin{align*} && k \tan \theta &= k \tan \alpha - \frac12 \frac{g}{V^2} \sec^2 \alpha k^2 \\ && 0 &= -k \tan \theta + k \tan \alpha -\frac12 \frac{gk^2}{V^2} (1 + \tan^2 \alpha) \\ && &= k \tan \theta + \frac12 \frac{gk^2}{V^2} - k \tan \alpha + \frac12 \frac{g k^2}{V^2} \tan^2 \alpha \end{align*} This is a quadratic in \(\tan \alpha\) which will (in general) have two solutions for \(\alpha\). Notice that if \(\tan z_1\) and \(\tan z_2\) are the two solutions then note that: \begin{align*} \tan z_1 + \tan z_2 &= \frac{k}{\frac12 \frac{g k^2}{V^2}} \\ &= \frac{2V^2}{gk} \\ \\ \tan z_1 \tan z_2 &= \frac{k \tan \theta}{\frac12 \frac{g k^2}{V^2}} + 1 \\ &= \frac{2V^2 \tan \theta}{gk} + 1 \end{align*} and so \begin{align*} && \tan (z_1 + z_2) &= \frac{\tan z_1 + \tan z_2}{1 - \tan z_1 \tan z_2} \\ &&&= \frac{\frac{2V^2}{gk} }{1 - \left ( \frac{2V^2 \tan \theta}{gk} + 1\right)} \\ &&&= -\frac{\frac{2V^2}{gk}}{\frac{2V^2 \tan \theta}{gk}} \\ &&&= - \cot \theta \\ &&&= \tan (\theta + 90^\circ) \end{align*} Therefore \(z_1 + z_2 = \theta + 90^\circ\) by considering physical options for the projectile. \begin{align*} && \cos \alpha &= \frac{k}{Vt} \\ && \sin \alpha &= \frac{k \tan \theta + \frac12 g t^2}{Vt} \\ \Rightarrow && 1 &= \frac{k^2}{V^2} \frac{1}{t^2} + \frac{k^2 \tan^2 \theta}{V^2 t^2} + gk \tan \theta + \frac14 \frac{g^2}{V^2} t^2 \\ \Rightarrow && 0 &= \frac{k^2}{V^2} (1 + \tan^2 \theta) + (gk \tan \theta -1)t^2 + \frac14 \frac{g^2}{V^2} t^4 \end{align*} This is a quadratic in \(t^2\), with the product of the roots being \(\frac{\frac{k^2}{V^2} (1 + \tan^2 \theta) }{ \frac14 \frac{g^2}{V^2}} = \frac{4}{g^2} (k^2 + k^2 \tan^2 \theta^2)\), therefore \(gT_1T_2 = g \sqrt{ \frac{4}{g^2} (k^2 + k^2 \tan^2 \theta^2)} = 2 \cdot OP\)
The ends \(P\), \(Q\) of a thin straight rod are constrained to move on two straight lines \(OX\), \(OY\) respectively that are perpendicular to each other. If the velocity of \(P\) is constant, prove that the acceleration of any point on the rod is at right angles to \(OX\), and find how it varies for different points of the rod.
A train of mass \(M\) is pulled by its engine against a constant resistance \(R\). The engine works at constant power equal to \(H\) units of work per second. Find the time taken for the velocity to be increased from \(v_0\) to \(v_1\) feet per second.
A smooth wedge of mass \(M\) and inclination \(\alpha\) (\(< 90^\circ\)) has one face in contact with a horizontal plane. A particle of mass \(2M\) is placed on the inclined face and allowed to slide down. Show that the horizontal acceleration of the wedge is \[g\sin 2\alpha/(2 - \cos 2\alpha),\] and find the force exerted on the table during the motion.
A uniform billiard ball of radius \(r\) is at rest on a rough horizontal table. The ball is struck a horizontal blow in a vertical plane through its centre at a height \(ir\) above the table, and the initial speed of the centre of the ball is \(V\). Find the linear velocity of the ball as the slipping ceases.
Two equal uniform circular discs are lying flat on a smooth horizontal table connected by a taut inextensible string to points \(A\) and \(B\), one on the edge of each disc. The string AB is a common tangent parallel to the line of centres. An impulsive couple is applied to one of the discs in a sense tending to stretch the string and of such magnitude that it would give the disc an angular velocity \(\omega\) if it were free. Find the actual angular velocity.