Hitting- Using the momentum provided by the pitcher
Catching- recieving the ball of great velocity
When a pitcher delivers a pitch, the ball is going to travel at a certain velocity. He begins to generate force from the lower half of his body allowing for the overall time he's holding the ball is longer, which leads to a greater change in momentum of the ball. The initial velocity when it leaves his hand is greater than when it gets to the plate.
The pitcher supplies the momentum for the batter apply an impact to the ball by swinging his bat to produce a change in momentum of the ball. Changing the momentum is known as impulse. They can maximize the effect force by swinging the heaviest bat they can, but swinging it just as fast as a lighter bat. Swinging the heaviest bat at the fastest speed possible while making contact with a pitch of higher velocity , combined with following through with their the swing for the longest time will cause the ball to travel the furthest. They follow through because it increases the time of impact the force is effecting the mass and velocity of the ball(momentum-produced from pitcher).
The catcher has to be strategic when receiving a pitch because it is coming in with a great velocity. The momentum of the ball was generated by the pitcher, so the catcher compensates and minimizes the effect of force from the ball on his hand. This is referred to as impulse, and in order to minimize the effect of force, the catcher moves their hand back as they receive the ball to increase the time of impact and decrease the force.