The physics
A rod moving across a magnetic field pushes the charges along itself. So it acts like a cell. Its emf is B v ℓ, where ℓ is the length that cuts the field.
ε = B v ℓ
Join the rails through a resistance R and a current flows. The field then pushes on that current, against the motion. This magnetic drag grows with the speed.
I = B v ℓ / R F = B I ℓ = B² ℓ² v / R
A loop makes an emf only while its flux changes. Inside a uniform field it moves freely with no emf. Its edges make an emf while it enters or leaves.
On upright rails a falling rod speeds up until the drag equals its weight. At that terminal speed, gravity's work all becomes heat.
m g = B² ℓ² vt / R