When the van moves at a steady speed on a straight horizontal road, that speed is $50\,\text{m s}^{-1}$. Show that $k = 0.5$, and determine the acceleration of the van when its speed is $25\,\text{m s}^{-1}$ on this straight horizontal road.
The van starts to climb a hill inclined at an angle $\theta^\circ$ to the horizontal. It travels along the line of greatest slope of the hill. At the start of the hill, the van’s speed is $20\,\text{m s}^{-1}$, and its acceleration is $5\,\text{m s}^{-2}$. Later, on the same hill, the van’s speed is $30\,\text{m s}^{-1}$, and its acceleration is $a\,\text{m s}^{-2}$. The power of the van’s engine stays at $62.5\,\text{kW}$, and the resistance force stays at $0.5v^2\,\text{N}$. Find the value of $a$ and the value of $\theta$.