In the sport of skeleton a participant jumps onto a sled (known as a skeleton) and proceeds to slide down an icy track, belly down and head first. In the 2010 Winter Olympics, the track had sixteen turns and dropped 126 m in elevation from top to bottom. (a) In the absence of nonconservative forces, such as friction and air resistance, what would be the speed of a rider at the bottom of the track? Assume that the speed at the beginning of the run is relatively small and can be ignored. (b) In reality, the gold-medal winner (Canadian Jon Montgomery) reached the bottom in one heat with a speed of 40.5 m/s (about 91 mi/h). How much work was done on him and his sled (assuming a total mass of 118 kg) by nonconservative forces during this heat?
To solve this problem, we need to apply the principles of conservation of mechanical energy and work-energy theorem.
(a) In the absence of nonconservative forces, the total mechanical energy of the rider at the top and bottom of the track will remain the same. We can calculate the potential energy at the top of the track (PE_start) and the kinetic energy at the bottom of the track (KE_end).
The potential energy at the top is given by the equation:
PE_start = m * g * h
Where:
m = mass of the rider and sled (118 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height of the track (126 m)
PE_start = 118 kg * 9.8 m/s^2 * 126 m = 146,724 J
Since there is no initial speed, all of the potential energy is converted into kinetic energy at the bottom. Therefore, the kinetic energy at the bottom is equal to the potential energy at the top:
KE_end = PE_start = 146,724 J
To calculate the speed at the bottom (v_end), we use the formula for kinetic energy:
KE_end = 0.5 * m * v_end^2
Rearranging the equation to solve for v_end:
v_end = sqrt((2 * KE_end) / m)
Substituting the known values:
v_end = sqrt((2 * 146,724 J) / 118 kg) ≈ 17.98 m/s
Therefore, in the absence of nonconservative forces, the speed of the rider at the bottom of the track would be approximately 17.98 m/s.
(b) In reality, nonconservative forces such as friction and air resistance are present, which cause energy loss in the system.
To calculate the work done by nonconservative forces, we use the work-energy theorem:
Work_done = KE_end - KE_start
Where:
KE_start = initial kinetic energy (which is zero because the initial speed is small and can be ignored)
KE_end = final kinetic energy (40.5 m/s in this case)
KE_start = 0 J
KE_end = 0.5 * m * v_end^2 = 0.5 * 118 kg * (40.5 m/s)^2 ≈ 96,788 J
Work_done = 96,788 J
Therefore, the total work done on the rider and sled by nonconservative forces during this heat is approximately 96,788 J.