We consider three components of energy loss: Rolling resistance, Form drag and Skin friction drag.
is a constant force opposing movement. Energy is lost to friction and hysteresis effects arising from the deformation of the wheel and the surface on which it rides
Cr is the rolling resistance coefficient. This is defined as the ratio of the force needed to roll the wheel to the force on the wheel. It depends on the type of wheel.
Er = energy in MJ to move against rolling resistance for 1km
Where M is the mass of the vehicle in kg
is the aerodynamic resistance caused by pushing air out of the way of a body. At the speeds and dimensions we are considering the flow is turbulent.
Ed = energy in MJ to move against form drag for 1km
Where Cd is the drag coefficient which is typically between 0.2 for a very streamlined shape and 1 for a flat surface.
A is the cross sectional area in m2
V is the velocity in msec-1
is the aerodynamic resistance due to pulling a long body with uniform cross section through the air without displacement.
L = length
Pr = length of outline of cross-section
Renolds number, 
is given by
For the velocity and dimensions we are considering Renolds number is large and the flow will be turbulent. The skin friction drag coefficient is given by:
The dynamic pressure:
The drag for the surface area 
is:
Where:
This would be for a smooth surface with no protuberances or gaps. Real surfaces, such as the outside of a high speed train, will have gaps between the coaches to allow articulation and projections such as wheels and power pick-up. To allow for this we have introduced
Rf = roughness coefficient. Has a value of 1 for a smooth surface and gets larger as the surface gets rougher.
Ef = energy in MJ needed to move against skin friction for one km:
Total Energy to move one cabin:
