Best explanation I've seen for why rail is superior when you want to move a lot of people or cargo:

[neroden]

The reason trains are better for moving large numbers of people or cargo than other methods is that trains can be very long and go quite fast around corners, without fishtailing (like "road trains") -- this means they can have very high throughput. But why can trains do that, and why can't buses or other "futuristic" gadgets like monorails do that?

Some of you will know this already, but -- it's the conical wheels.

The reason trains can be very long, and go quite fast around corners, without fishtailing or overturning, with no complicated or expensive active stabilization -- is the passive stabilizing properties of the conical wheels. The physics is explained here, better than I've ever seen it explained before:

The interesting engineering behind the SHAPE of Train wheels! - YouTube

Most people don't know this. I think it's important to communicate as part of advocacy. It's the key reason why buses, or trucks, or monorails, or gadgetbahns, are just not as good.

 

[George Harris]

Somewhat nice, but somewhat simplistic introduction to a part of the equation, but then I have been involved professionally with this sort of stuff for most of the last 50 years. Do pay particular attention to what happens when you reverse the direction of the taper. There are numerous studies, full scale, modeling, and computer involved in development of the taper and what it should be. There have been full scale standards tried for multiple years by many railroads and transit systems around the world over the last 140+ years. The wheel shape is only part of the equation. The shape of the rail head is another significant part of it. Wheels on rails have been reality for nearly 190 years by now, but what appears to be a consensus only goes back about 25 years. Even that is not generally accepted by all systems, as there are some that consider themselves, rightly or wrongly, as "smarter than the average bear." Actual speed on curves is not limited by the wheel shape, but by centrifugal force, and particularly for passenger systems, comfort.

Wheel shape is only a small part of the advantage of rail mounted freight or passenger trains over road mounted equipment. Most of the others, such as "monorail" or maglev suffer from other issues, with technological complexity being significant. The massive cost and complexity of any form of turnout into, out of, or between tracks with these systems is also a major show stopper. With road equipment, the inability of multiple following units to follow the path of the leading unit is the killer.

For wheels, the taper is usually 1 in 20, with two or more differing and decreasing radii going toward the flange. These smaller radii improve centering on smaller radius curves. This multi-radius cross slope is commonly referred to as a "worn wheel" profile, but that statement is not truly accurate, as there are some issues that develop as wheels wear that you do not want. However, it does reduce the rate of wheel tread wear significantly.

For rails, the current conclusion is that the preferred crown radius is 8 inches, or 200 mm in the metric world. (200 mm is for all practical purposes 8 inches.) Interestingly enough, these values were derived independently by several studies and experiments under a wide variety of traffic types, without the two major experimenters having knowledge of the other's work. Again, there is a smaller shoulder radius, or two, followed by a corner radius usually in the 1/2 inch + range. All currently recommended AREMA rail sections have this 8 inch crown radius, and the most common European section, EN 60, formerly called UIC 60 has the 200 mm crown radius. This 200 mm crown radius is a revision from the 300 mm crown radius which was its standard for many years. Likewise, the 8 inch crown or AREMA rails is a revision from the 10 inch crown that was standard on the most common sections for many years. Further, rails are not normally set vertically. They are inclined toward the center, most commonly at 1:40 in the US and quite a few other places, and with greater slopes, usually 1:30 to 1:20 by some users. They are still installed vertically in turnouts, so there is a twist off the ends of every turnout. With the 8 inch crown radius, this is of no significance.

 

[TrackWalker]

We have curves. Not corners.