In home-run contest you will see how far you are able to throw the sandbag. There are four differences from the home-run contest in SSBM:
After the game begins, attack the sandbag to get it's damage percentage up as high as you can in 10 seconds. Before the timer runs out, grab the bat, get close to the sandbag and do a
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smash to send it flying. The camera will scroll to follow the sandbag's flight. Don't worry if it goes off the top of the screen; this can be an indicator that it will stay in the air for a long time. Once it hits the ground, it will bounce a few times and then skid across the ground before coming to a stop. Once it stops, it will show you, in feet or meters (depending on the language you're using for the game), how far you hit it. This is your score. After this, the game will prompt you to restart, go back, or watch a replay.
There are many strategies for Home Run contest. You can only hit the sandbag against the shield surrounding the platform a few times before the shield breaks, so it's hard to figure out how to deal maximum damage while not letting sandbag hit the shield too many times. One strategy involves throwing the bat at the sandbag, which can deal decent damage at a range. You can usually throw the bat twice at the start before changing to standard attacks, and sandbag will be right beside you. As a final tip, hitting with the inside of the bat causes the sandbag to fly higher but not as far, but hitting with the tip will give you maximum distance, so placing yourself farthest away from the sandbag before performing the final smash attack is key. It's risky, but if you can successfully connect with the sandbag you will get a lot of extra distance.
On a side note, the Smash Bros. Dojo states in their Home-Run records that a pair of Ganondorfs managed to launch the Sandbag over 15,000 feet, thus being the world record for co-op matches.