The balloon is going up. Luckily you've attached a line to it in order to stop it.
Usages of stop
You stop next to a small dog and ask him what age he is.
止む is used when something stops on its own. It's often used for weather phenomena, such as rain or snow.
止まる is a broader term that can be used for any sort of cessation of movement or action.
Your tooth will stop functioning if you eat a container of rice.