On the right we see a train negotiating a curve to the left.  Superimposed are the unbalanced forces that tend to tip the train toward the outside of the curve, much as in the Circling: Ground and Sky puzzle.

Figure 1 is a sketch of a train-car traveling along a curve of radius R on a track with gauge G.  Four forces are shown acting on the car.  The support provided by the inside and outside rails are Fi and Fo respectively.  Weight W is shown acting vertically from the vehicle's center-of-mass, which is located at height H above the track.  Centrifugal force Fc acts on the train-car horizontally away from the track's center of curvature.

Sharp curves are out of the question.  For speed, radius of curvature R needs to be made as long as possible, but there are often limitations imposed by terrain or land-use commitments. 

Figure 2 shows how superelevation, symbolized by E, can be used much like a banked highway to balance the rail forces (Fi = Fo ) thereby allowing trains to operate faster than might otherwise be possible along curvatures in the trackway. 

For given values of R and E, the two track forces can be balanced at only one particular train train speed, VBALANCED...

Fc / W = E / G, and...
Fc = (W / g R) (VBALANCED)², so that...
VBALANCED = [(E / G) (g R)]^1/2

Figure 3 is included here to address another limit, this time on the values of superelevation E in view of gravity's tendency to tip a stopped train toward the inside of the curve.  The effect is worsened by the draw-forces between vehicles as the train starts up. 

Every segment of trackway must be certified with what is called a 'civil limit', which specifies the maximum speed that a train can operate on it. 

In additon to the dimensions of curves as indicated here, civil limits must take into consideration all trackway conditions, including grades, switches, and station platforms.

Superelevations are generally specified for an intentional 'unbalance' of forces as shown in Figure 4.  Thus, at speeds approaching civil limits, the train will tend to be tilted outward from the center of curvature. 

Of course, trackways are never designed to change abruptly from straight to curved and back again.  Instead, the rails are given a 'spiral' shape, gradually changing radius of curvature, between infinitely long, for straight trackway, to a fixed radius R, for a circular segment in the 'alignment'. 

Along the spiral, superelevation changes linearly from zero to its value E as indicated in Figure 4.

Meanwhile, objects inside the train -- in particular passengers -- will be acted upon by invisible, gradually changing forces from the left or the right every time the train enters a curve from straight, tangent track. 

The tilting train was invented to overcome discomfort for passengers, the first being put into service in 1938.  Figure 5 is a sketch of a high-speed passenger car traveling in a curve and tilted by a mechanical device atop the bogie.  The tilt angle has been adjusted on the fly so that the passenger apparently experiences no lateral forces.  Complete balance might seem to be ideal.

It is not.

According to an article published by the Locomotive & Carriage Institution, tilt technology is used primarily for passenger comfort not speed; however, 'tilt nausea’ can result from compensating out lateral forces altogether.

Figure 6 shows the preferred application.  An accelerometer fitted in the first bogie of the train measures lateral forces as the train enters a curve.  The signal is processed by computer and sent down the trainline.  Computer-controlled hydraulic rams tilt each coach into the curve, up to a maximum inclination of 6.5º.  The tilting system compensates for no more than 75% of the lateral force of a curve and is only employed at speeds above 44 mph.  Incidents of tilt-train 'sea-sickness' are rare, as 25% of lateral forces are still felt by the passengers.

High speed trains require great distances to slow down for an upcoming curve and to speed back up again when the track ahead is straight.  Best to design curves so that slowing will not be necessary.  Given the limitations described above, that may not always be possible.  Solvers of the Fast Track puzzle are invited to consider the following case:

Cruise Speed ~~~~~~~~~~~~~~~~ VCRUISE = 200 mph
Track Gauge ~~~~~~~~~~~~~~~~~~G = 1 metrum asinorum
Height of Center of Mass ~~~~~~~~~~ H = 1¹/2 G
Superelevation ~~~~~~~~~~~~~~~~~~~ E = 7 in
Unbalance ~~~~~~~~~~~~~~~~~~~~~~~~U = 4 in
Radius of Curvature ~~~~~~~~~~~~~~~~~~ R = 1 mile
Change in Alignment ~~~~~~~~~~~~~~~~~~~ A = 20º 
Spirals (in + out) ~~~~~~~~~~~~~~~~~~~~~~~~ S = A / 2
Acceleration/Deceleration ~~~~~~~~~~~~~~~~~~~ a = d = g / 10
Car Length ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C = 75 ft
Maximum Train Length ~~~~~~~~~~~~~~~~~~~~~~~ L = 20 x C = 1,500 ft