Chapter 5, Problem 66TP

Dangerous Ride. You and your exploration team are stuck on a steep slope in the Andes Mountains in Argentina. A deadly winter storm is approaching and you must get down the mountain before the storm hits. Your path leads you around an extremely slippery, horizontal curve with a diameter of $90 \text{m}$ and banked at an angle of $40.0°$ relative to the horizontal. You get the idea to unpack the toboggan that you have been using to haul supplies, load your team upon it, and ride it down the mountain to get enough speed to get around the banked curve. You must be extremely careful, however, not to slide down the bank: At the bottom of the curve is a steep cliff.

1. Neglecting friction and air resistance, what must be the speed of your toboggan in order to get around the curve without sliding up or down its bank? Express your answer in $\text{m}/\text{s}$ and $\text{m}.\text{p}.\text{h}.$
2. You will need to climb up the mountain and ride the toboggan down in order to attain the speed you need to safely navigate the curve (from part (a)). The mountain slope leading into the curve is at an angle of $30.0°$ relative to the horizontal, and the coefficient of kinetic friction between the toboggan and the surface of the slope is $\left({\mu }_{\text{mountain }}=0.15\right).$ How far up the mountain (distance along the slope, not elevation) from the curve should you start your ride? Note: the path down the mountain levels off at the bottom so that the toboggan enters the curve moving in the horizontal plane (i.e., in the same plane as the curve).