The physics
A moving body carries kinetic energy. It grows with the mass and with the square of the speed.
K = ½ m v² v = √(2K / m)
A steady push F over a distance d does work Fd. On a smooth strip, all of that work becomes kinetic energy. To go from u to v, a push must give the change in KE.
F d = ΔK = ½ m (v² − u²)
KE and momentum go together. A push F for a time t gives any cart the same momentum, p = Ft. For the same p, a heavier cart has less KE.
p = m v K = p² / 2m
Brakes take the KE away. Their work, F × d, equals the KE. Two cars with the same KE and brakes stop in the same distance.
d = K / F