it can be complicated but you either need to put the cart on a slope and get in quickly, or you need to make power rails and put redstone torches next to them.
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To move in a mine-cart, you have to have to have it moving before you get in. If you get in without it moving, you cannot get it to move without a booster rail and a button. The best way to move in a minecart is to push the minecart in front of you then jump in just before it goes down a hill.
to play a map, open minecraft, then click singleplayer, choose the world, then click play selected world.
Simply drag you old world maps from your old "saves" folder to the new "saves" folder. Application Support/ minecraft (or .minecraft for Windows) /Saves.
right click and click "move to trash" or "delete"
you have to go to start, click run, click your .minecraft folder, take the downloaded mod (it has to be zipped) click open archive, then with the .minecraft folder, you find bin go in, and open minecraft's archive, then you move the files in the mod archive folder to the minecraft archives. (some things might be wrong, but that's because i used pure memory to answer this, and i have a bad memory. )