A rain-drop falls through air containing stationary infinitesimal water droplets. The volume-concentration of droplets is \(c\) (that is, they occupy a fraction \(c\) of the unit of space). The rain-drop maintains its spherical shape during the motion, and coalesces with all the droplets in its path. Show that, at time \(t\), its velocity \(v\) and radius \(r\) satisfy the equations \begin{equation*} \frac{dr}{dt} = \frac{cv}{4}, \quad \frac{d^2}{dt^2}r^4 = cgr^3. \end{equation*} The drop starts from rest as an infinitesimal droplet. Assuming that its acceleration is uniform, show that it is \(g/7\).
By writing \(x = r\cos\theta\) and \(y = r\sin\theta\) (where \(r\), \(\theta\) are polar coordinates at origin \(O\)), or otherwise, show that the components of acceleration of a particle \(P\) along and perpendicular to \(OP\) are \begin{equation*} \ddot{r}-r\dot{\theta}^2 \quad \text{and} \quad r\ddot{\theta}+2\dot{r}\dot{\theta} \end{equation*} respectively, where dots denote differentiation with respect to time. A particle of unit mass is moving under the action of a force \(F(1/r)\) directed toward the origin. Show that \begin{equation*} r^2\dot{\theta} = h, \end{equation*} where \(h\) is a constant, and also that, if \(u = 1/r\), \begin{equation*} h^2u^2\left(u+\frac{d^2u}{d\theta^2}\right) = F(u). \end{equation*} Find the equation of the path of the particle if \(F(u) = Au^3\), where \(A < h^2\).
An aircraft is flying above a plane inclined at an angle \(\alpha\) to the horizontal. A smooth sphere is dropped from the aircraft when it is travelling horizontally with speed \(u\) and at such a height as to make the sphere impinge normally on the plane. Show that the sphere travels a distance \begin{equation*} \frac{2u^2e^2}{g\sin\alpha\cos^2\alpha(1-e)^2} \end{equation*} along the plane before it ceases to bounce. (Here \(e\) is the coefficient of restitution between the sphere and the plane.)
State the laws of conservation of linear momentum and energy for the motion and collision of perfectly elastic smooth spherical particles referred to a fixed frame of reference \(F\) in the absence of external forces. A second frame of reference \(F'\) moves with velocity \(\mathbf{v}+\mathbf{a}t\) relative to \(F\) where \(\mathbf{v}\) and \(\mathbf{a}\) are constant vectors and \(t\) is time. Prove that the conservation laws holding in \(F\) hold also in \(F'\) if and only if \(\mathbf{a} = \mathbf{0}\). What alternative equations of motion hold in \(F'\) if \(\mathbf{a} \neq \mathbf{0}\)?
Three linear springs each of modulus \(\lambda\) and natural length \(l\) are connected end to end and lie in a straight line on a smooth horizontal table. At each of the two points where the springs join, a mass \(m\) which is free to move is attached. The two ends of the composite spring are attached to the table, so that in equilibrium the springs are all stretched. If \(x\) and \(y\) denote small displacements of the masses from their equilibrium positions along the line of the springs, show that \begin{equation*} ml(\ddot{x}+\ddot{y})+\lambda(x+y) = 0 \end{equation*} and \begin{equation*} ml(\ddot{x}-\ddot{y})+3\lambda(x-y) = 0. \end{equation*} Describe exactly the subsequent motion if, at \(t = 0\), one of the masses is given a sudden unit velocity towards the second which is itself stationary.