Physics Question, What formula would be used?

Question

Why can a batter hit a pitched baseball further than a ball tossed in the air by the batter?

P=mv?

Answer 1

The batter does provide an impulse, or change in momentum, to the ball but this is not the explanation for what you were describing.

The impulse equation is: change in momentum = mass * change in velocity = average force times change in time. In variable form J = m*del_v = F*del_t.

This explains why the velocity changes but not why a larger initial velocity facilitates a larger final velocity.

Based on that alone you might think that a slower pitched ball would fly further.

The reason that is NOT the case is due to conservation of energy. A ball thrown very hard has a certain amount of kinetic energy and that energy has nowhere to go. When the bat strikes it, as long as the bat doesn't absorb too much energy in the form of vibrations, the ball will keep the energy and also gain some energy from the batter. That's why the ball is hit harder when it strikes the "sweet spot".