A shell of mass \(M\) is fired vertically into the air from ground level, and is given an initial kinetic energy \(E\). It explodes at a height \(h\), dividing into two equal fragments. The energy generated in the explosion is \(\frac{1}{2}E\), and it is wholly converted into kinetic energy of the fragments. Show that if air resistance is negligible both fragments reach the ground at times no later than \[T = \frac{3}{g}\sqrt{\frac{2E}{M}}\] after the shell was fired, where \(g\) is the acceleration due to gravity.
A rocket is launched vertically from rest against a constant gravitational acceleration \(g\). The fuel is burnt at a uniform rate in a time \(T\) and is ejected at a constant speed \(u\) relative to the rocket. Initially \(\frac{3}{4}\) of the total mass is fuel. Show that the maximum upward velocity achieved by the rocket is \[2u\ln 2 - gT\] provided \[T < 3u/4g.\] What is the significance of this condition? Show also that if \(T < 3u/4g\), the maximum height attained is \[2(u\ln 2)^2/g - \left(\frac{8}{3}\ln 2 - 1\right)uT.\]
A particle at position \(\mathbf{r}(t)\) is subject to a force \(\mathbf{E} + \dot{\mathbf{r}} \times \mathbf{H}\) per unit mass, where \(\mathbf{E}\) and \(\mathbf{H}\) are constant unit vectors and \(\mathbf{E} \times \mathbf{H} \neq \mathbf{0}\). If \[\mathbf{r} = \alpha\mathbf{E} + \beta\mathbf{H} + \gamma\mathbf{E} \times \mathbf{H},\] derive the differential equations that must be satisfied by \(\alpha(t)\), \(\beta(t)\) and \(\gamma(t)\). The particle starts from the origin at time \(t = 0\) with \(\dot{\alpha} = \dot{\beta} = \dot{\gamma} = 1\). Show that the subsequent motion is given by \begin{align*} \alpha &= \sin t\\ \beta &= (1 + \mu)t + \frac{1}{2}\mu t^2 - \mu\sin t\\ \gamma &= 1 + t - \cos t \end{align*} where \[\mu = \mathbf{E}.\mathbf{H}.\]
Three identical spheres of radius \(a\) and mass \(m_1\) are touching on a horizontal table. The coefficient of friction between each sphere and the table is \(\mu_T\). A fourth sphere of radius \(b\) and mass \(m_2\) rests on the first three spheres. The coefficient of friction between the fourth sphere and the other three is \(\mu_S\). Show that if there is no slipping, \[\mu_S > \tan \tfrac{1}{2}\theta\] and \[\mu_T > \frac{\tan \tfrac{1}{2}\theta}{(3m_1/m_2) + 1}\] where \[\sin \theta = \frac{2a}{(a+b)\sqrt{3}}.\]
An elastic string is held between two fixed supports P, Q which are a distance \(3d\) apart. The tension in the string is proportional to its extension, and is \(k^2md\) when the length of the string is \(3d\). A bead of mass \(m\) is attached to the string a distance \(d\) from P, and an identical bead is attached a distance \(d\) from Q. Find the equations of motion of small displacements \(\alpha\), \(\beta\) of the beads perpendicular to PQ. Ignore gravity and motion parallel to PQ. Show that \(\alpha + \beta\) and \(\alpha - \beta\) each undergo simple harmonic motion, and find the periods. Describe the motions corresponding to \(\alpha + \beta = 0\), and \(\alpha - \beta = 0\). At time \(t = 0\) \[\alpha = \alpha_0, \beta = \beta_0, \frac{d\alpha}{dt} = \frac{d\beta}{dt} = \gamma_0.\] Show that if \[\frac{(\alpha_0 + \beta_0)k}{2\gamma_0} = -\tan\left(\frac{\pi}{2\sqrt{3}}\right)\] then \(\alpha = \beta = 0\) at some subsequent time.
Two uniform rods AB, BC, of lengths \(2a\) and \(2b\) and masses \(m_1\) and \(m_2\) respectively, are smoothly jointed at B. They lie in a straight line on a smooth horizontal table. A horizontal impulse \(F\) is applied at A, perpendicular to AB. Find the initial velocity of C, and show that the kinetic energy \(T\) generated is \[T = \frac{F^2(4m_1 + 3m_2)}{2m_1(m_1 + m_2)}.\] If instead the impulse were applied at C, what would be the initial velocity of A?