The Energy = Int{e*i*dt} = (1/R)*Int{(dPhi/dt)^2*dt}
Now this integral is the area under the function dPhi/dt^2 vs. time.
To see if this can be velocity dependent take the simple, crude model in
which Phi(t) is piecewise linear and triangular; ie. Phi(t) rises linearly
from zero to a maximum value (P) in a time T/2 seconds, and then falls
linearly back to zero in another T/2 seconds. Then dPhi/dt^2 has the
constant value (2*P/T)^2 for T seconds, and is zero otherwise.
Thus the Energy = (4*P^2)/(R*T) and is inversely proportional to T, the time
of travel (or directly proportional to the velocity).