力学

A particle of mass $m$ is projected vertically upwards from the ground with initial speed $u$. The particle experiences a drag force of magnitude $mkv^2$ where $v$ is its speed and $k > 0$ is a constant.

Show that while the particle is moving upwards, $v\dfrac{dv}{dx} = -g - kv^2$, where $x$ is the height above the ground. Hence find the maximum height $H$ reached.

Write the equation of motion for the particle as it falls from height $H$. Find the speed $V$ when it returns to the ground.

Show that $V < u$ and explain this physically.

Find the limit of $V$ as $k\to0$, and describe the behaviour of $V$ as $k\to\infty$.