Just for a bit of fun perspective, that’s 7 km/s faster than reentry velocity! Even if you felt like dealing with the heat, it would cost an ungodly amount of energy which kind of defeats the point of an elevator. On the other hand, you could ignore radiation shielding if the trip is short enough? Not worth the trade off I think.

Wouldn’t you hit that velocity far above the troposphere, where there’s much less air resistance?

Not sure if this would work, but how about using a wire with evenly-spaced metal beads, then using a magnet (a large one like used in MRI machines) to pull up the cart? You'll probably have to turn the magnet on/off at a specific frequency, depending on the momentum of the cart, which could be a challenge.

power = force * distance / time

If you want constant acceleration, you need to maintain constant force. A higher-velocity vehicle applies that force across a greater distance per unit time, and therefore requires more power.

Accelerating 1 metric ton at 1g requires 9800 N. So at peak velocity (18 km/s), that's 180 megawatts per ton, which is more than 700 times the power to mass ratio of a Tesla P100D.