A 3 kg cart at the bottom of a hill is trying to move up a frictionless hill that is 22 meters tall?

You are talking about conversion of kinetic energy to potential energy.

PE = mgh = 3kg x 9.81m/s^2 x 22m = 647.46(m^2/s^2)-kg
KE = (1/2)mv^2 = 1.5kg x v^2

KE = PE

1.5kg x v^2 = 647.46(m^2/s^2)-kg The rule is you can do any valid operation on both sides of an equation and it will still be equal. Divide by 1.5kg.

v^2 = 431.64(m^2/s^2) Square root.

v = 20.776m/s <-- ANSWER