The reason that sharks swim with their mouths open is so the shark can breathe underwater while swimming.
The shark has their mouth open while swimming to allow the water to pass over their gills which allows the shark to breathe.
If the shark had it's mouth closed they would likely not be able to breathe and would die so they keep their mouths open while swimming.
Most sharks can swim around 50 miles per day which is quite the distance.
However some sharks are thought to be able to swim as far as 60 miles but most sharks swim at most 50 miles or less per day.
The shark will only swim farther off than they currently are if they need too.
Otherwise the shark may just stay close to where they are if there is food in that area.
If the shark needs to find more food then they may decide to swim off to look for more food in the waters.
But if the shark has an abundance of food to feed off in their current location then they will usually stay close and only swim off a farther distance if they need too.
Most sharks prefer waters with sea surface temperatures of 50 to 72 degrees Fahrenheit and they can usually swim as deep as 6,150 feet deep in the waters.
Some certain types of sharks including the nurse sharks, angel sharks and lemon sharks have strong neck muscles that pump water over the gills, allowing the sharks to breathe while resting on the ocean bottom.