1 problem found
Two identical particles of unit mass move under gravity in a medium for which the magnitude of the retarding force on a particle is \(k\) times its speed. The first particle is allowed to fall from rest at a point \(A\) whilst, at the same time, the second is projected upwards with speed \(u\) from a point \(B\) a positive distance \(d\) vertically above \(A\). Find their distance apart after a time \(t\) and show that this distance tends to the value \[ d+\frac{u}{k} \] as \(t\rightarrow\infty.\)
Solution: Both particles have equations of motion, \(\ddot{x} = -g-k\dot{x}\), so we can note that the distance between them has the equation of motion: \(\ddot{x} = -k \ddot{x} \Rightarrow x = Ae^{-kt} + B\) \begin{align*} && x(0) &= d \\ \Rightarrow && A+B &= d \\ && x'(0) &= u \\ \Rightarrow && -kA &= u \\ \Rightarrow && A &= -\frac{u}{k} \\ \Rightarrow && B &= d+\frac{u}{k} \\ \Rightarrow && x(t) &= -\frac{u}{k}e^{-kt} + d + \frac{u}{k} \to d + \frac{u}{k} \end{align*} as required.