Your calculations are correct, with the caveat that that's the average power assuming constant acceleration, not the maximum power. But you can't launch a one-kilogram thing and get it through Earth's atmosphere; you need an aerodynamic fairing and ablative tiles and whatnot, not to mention whatever kind of inductive apparatus you're using to grab hold of it in the first place. Thus my remark about the sparrow!

Until your objects get long and thin enough that drag matters, Newton's impact depth approximation applies to the atmosphere: if you're going straight up, you have roughly 10 g of air per cross-sectional mm² that you will run into on the way, assuming you can keep the hypersonic aerodynamics sufficiently under control to keep from just totally tumbling end over end, which is harder than it sounds. At the shallow angles available running up mountainsides, the situation is several times worse.

It's unfortunate that people incapable of doing calculations themselves are downvoting you. Calculations like these, plus experiments to validate them, are how we got rockets in the first place.