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Part A: The Ski Lift (7.5 marks)
A ski lift is installed at a ski resort to pull skiers up a slope (Figure 1 below). The slope has a constant inclination of α=7°, a length L=930 m and the friction coefficient between the snow and the skis is μ=0.04. When lifting a skier, the bar connecting the skier to the aerial steel rope remains inclined by β=26° (with respect to the normal to the slope) and is subject to a time-varying tension T. For this exercise consider a skier of mass m=80 kg.
Figure 1 – Ski Lift Schematic
- Draw a FBD of the skier during the lifting process. Ensure you include all forces and a coordinate system. (1 mark)
- Obtain the equation of motion of the skier as a function of the only unknowns x (and/or its derivatives) and T (ie express all other quantities numerically). (1.5 marks)
- The skier starts their journey up the ski lift at t = 0, and they reach the top of the slope at tend=224 s. The skier’s position, x, as a function of time, t, over this time period is given by:
xt=L2ttend
Obtain the analytical expression of the tension with respect to time and compute the maximum tension value.
Note: The coordinate x is parallel to the slope (in the same direction as the motion of the skier) as marked in the diagram above. (2 marks)
- Determine an equation for the power exerted from the bar onto the skier with respect to time. (1 mark)
- Using the power equation, determine how much energy is imparted from the bar to the skier during the lift. (1 mark)
- The skier then reaches the top of the peak and proceeds to ski down the slope as per the image below.
Figure 2 – Ski Slope
The slope has a length of 1.3 km and is at an incline of θ=5°. The skier starts at rest at the top of the slope. What coefficient of friction is required between the skis and the snow to bring the skier to rest at the bottom of the slope? You may assume that the kinetic and static coefficients of friction are the same and the skier creates the same level of friction between their skis and the snow the whole way down. (1 mark)
Part B: Bumper Cars (7.5 marks)
Consider the bumper car collision below. Bumper car A is waiting to come out onto the circuit so it is stationary before the collision (Va=0). Bumper car B comes around the corner and decides to smash into bumper car A. The velocity of bumper car B at the instant before the collision is Vb=12 km/h and they collide at an angle α=25°. At the time of the collision bumper car A is in contact with the walls and can only move backwards. The bumper cars individually each have a weight of 150 kg however, the passenger in bumper car B weighs 110 kg and the passenger in bumper car A weighs 75 kg. For this collision you may assume that e=0.71.
Figure 3 – Bumper Car Collision
- Draw individual FBDs of both bumper cars during the collision. Ensure you show all relevant components and a coordinate system (1.5 mark)
Determine the velocities of each car the instant after the collision (report your solution as a vector in your chosen coordinate system). What is the impulse experienced by the wall? (6 marks)