When the batter hits the ball, that also counts as an unbalanced force.
Ball will move until hit or caught
When the catcher caches the ball, that would be the unbalanced force which causes the ball to stay at rest.
The basketball will not go as far as the baseball will because it has a bigger mass
Newtons Second Law of Motion/ Acceleration
The baseball will go further because it has a smaller mass.
Newtons Third Law of Motion/ Action and reaction
Newtons first law of motion tells us than an object at rest, will remain at rest until acted upon by an unbalanced force and the same for an object in uniform-motion. This can be seen in baseball because when you are pitching the ball and you throw it hard in a straight direction, the ball will stay in motion until it is either hit by the batter or caught by the catcher.
Newtons second law of motion tells us that the acceleration of an object is dependent on forces acting on the object and the mass of an object. This can be seen in baseball because if you are hitting the baseball, that baseball would go much farther than if you had hit a basketball. This happens because the basketball has a much bigger mass than the baseball does.
Newtons third law of motion tells us that for every action, there is an equal and opposite reaction. This can be seen in baseball because when you are hitting the ball, the ball makes contact with the bat which makes the bat go to the left but, that also means that when you are hitting, the ball makes the bat go to the right at the same time.