A particle \(A\) of mass \(m\) and a particle \(B\) of mass \(2m\) are connected by a light string of length \(a\) and slide on a smooth horizontal table. Initially both are at rest with the string taut, when another particle of mass \(m\) moving with velocity \(U\) perpendicular to \(AB\) embeds itself in \(A\). Show that \(A\) comes to rest again after a time \(\frac{2ma}{U}\). What is then the velocity of \(B\)?
The motion of a boomerang is illustrated by a particle of mass \(m\) moving in a horizontal plane with instantaneous speed \(v\) under the action of a tangential resistive force \(mkv^2 \cos \alpha\) and a normal force \(mkv^2 \sin \alpha\) tending to deflect the particle to the right, where \(\alpha\) is a constant acute angle. What is the shape of its path? If it is projected with speed \(v_1\), show that it returns to the point of projection after a time $$\frac{e^{2\pi \cot \alpha} - 1}{kU \cos \alpha}.$$
Find in terms of three non-zero vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), (such that \(\mathbf{a}\) is not perpendicular to \(\mathbf{b}\)) the most general vector \(\mathbf{r}\) which satisfies $$\mathbf{a} \times (\mathbf{b} \times (\mathbf{c} \times \mathbf{r})) = \mathbf{0},$$ examining carefully any configurations which give rise to exceptional cases.