I’m having some trouble with a worksheet I was given for undergraduate mechanics. I didn’t have any lectures in this topic due to strikes happening at my university, so any help with how to answer part b) would be hugely appreciated.
(edit) This is what I got for part a): my answer Thanks in advance
Under the announced forces the movement reads
$$ -m g \hat k + m \lambda \hat i - m \mu\dot z \hat k = m(\ddot x\hat i + \ddot z \hat k) $$
or
$$ \left\{ \begin{array}{rcl} \ddot x & = & \lambda \\ \ddot z +\mu \dot z + g & = & 0\\ \end{array} \right. $$
Now solving for the velocities
$$ \left\{ \begin{array}{rcl} v_ x & = & \lambda t + v_{x_0}\\ v_ z & = & -g/\mu + v_{z_0}e^{-\mu t}\\ \end{array} \right. $$
If the elapsed time is such that $e^{-\mu t} \ll 1$ then $\vert{v_z} \vert\approx \vert v_{z_\infty} \vert= g/\mu$.
Here
$$ \vert v\vert\ = \sqrt{v_x^2+v_{z_\infty}^2} $$