a) If you are at the top of a toboggan run that is \(40.0 \mathrm{~m}\) high, how fast will you be going at the bottom, provided you can ignore friction between the sled and the track? b) Does the steepness of the run affect how fast you will be going at the bottom? c) If you do not ignore the small friction force, does the steepness of the track affect the value of the speed at the bottom?

To find the speed at the bottom of the toboggan run without friction, we can apply the conservation of mechanical energy. In this case, the potential energy at the top of the run is converted into kinetic energy at the bottom. The formula for the potential energy is \(PE = mgh\), where \(m\) is the mass, \(g\) is the gravitational acceleration (\(9.8 \mathrm{~m/s^2}\)) and \(h\) is the height. At the bottom of the run, all the potential energy is converted into kinetic energy, which is given by \(KE = \frac{1}{2}mv^2\), where \(v\) is the speed at the bottom. Since the initial potential energy is equal to the final kinetic energy, we have: $$mgh = \frac{1}{2}mv^2$$ The mass \(m\) of the toboggan and rider is the same on both sides, so we can simplify the equation and solve for the speed \(v\) at the bottom: $$gh = \frac{1}{2}v^2$$ $$v^2 = 2gh$$ $$v = \sqrt{2gh}$$ Substituting the known values: $$v = \sqrt{2(9.8)(40.0)}$$

As we have found the final speed using the conservation of energy principle, we can see that it only depends on the initial height (\(h\)) and the gravitational acceleration (\(g\)), but not on the steepness of the run. Therefore, the steepness of the run does not affect the final speed at the bottom of the toboggan run when ignoring the friction between the sled and the track.

If we introduce the effect of friction, the steepness of the track has a significant impact on the final speed. When friction is present, the toboggan loses some of its mechanical energy during the descent. The steeper the track, the less contact between the sled and the track occurs during the descent, leading to less energy being lost to friction. Consequently, a steeper track results in a higher final speed at the bottom of the toboggan run compared to a less steep track, assuming friction exists between the sled and the track.