Final answer:
To determine the braking force of the cable car, we evaluate the gravitational force components on both the cable car and counterweight and ensure a net force of zero for constant speed. If brakes fail, the runaway speed is found using energy conservation, considering the conversion of potential energy into kinetic energy as the car travels down the hill.
Step-by-step explanation:
Understanding the Cable Car Braking Force and Runaway Speed
To calculate the braking force the cable car needs to descend at constant speed, we analyze the forces acting on the system which includes the gravitational force components of both the cable car and the counterweight. Assuming friction is negligible and there is no net acceleration, the total force across the system is zero when moving at constant speed. We'll consider the forces due to gravity that act along the slope and find the difference to get the required braking force.
For the second part of the question, when the brakes fail, the cable car will accelerate due to the net force which is the component of gravitational force along the slope. Using energy conservation principles, we can calculate the final speed of the cable car at the bottom of the hill.This problem involves concepts such as mechanical energy conservation, forces on inclines, and Newton's laws of motion.