Well, if we ignore the force being exerted on the bat by the batter, then we can use the principle of conservation of momentum, and if we make-believe that the collision is perfectly elastic, then we can say that kinetic energy is conserved as well. This gives us two equations to play with:
where the vs are initial velocities, ms are masses, and ws are subsequent (after-collision) velocities. Algebra then lets us derive two equations for the subsequent velocity of the ball in terms of the masses of the bat and ball, the subsequent and initial velocities of the bat, and the initial velocity of the ball:
and
In both these equations [edit: this is wrong; see addendum below], the subsequent velocity of the ball is an increasing function of the initial velocity of the ball (when all the other variables are treated as constants). Clearly, balls that are hit faster will travel farther, so this would seem to suggest that the faster pitch will be hit farther. However, in addition to the simplifying assumptions we made at the beginning, this analysis also assumes that the after-collision velocity of the bat is independent of the speed of the pitch, which is surely not true, so this reasoning cannot be said to have definitively resolved the question.
EDITED TO ADD: Except velocity and momentum are vector quantities, so the initial velocity of the ball should be regarded as having sign opposite of that of the initial velocity of the bat. This would seem to mean that our momentum equation is actually saying that the slower pitch will be hit further … I’m confused.
This would seem to mean that our momentum equation is actually saying that the slower pitch will be hit further … I’m confused.
No reason to be confused, remember you have ignored the force exerted by the batter. If the ball is enough heavy and fast, it will simply shoot away the bat and continue forward. (Edit: the mistake you made is
(when all the other variables are treated as constants)
which you can’t, since the posterior bat velocity isn’t independent of the prior ball velocity.)
Let’s make it explicit:
Let w and W are velocities of the ball and the bat after the hit, v and V are the velocities before the hit, m and M are masses of the ball and the bat, respectively. Then we have
v+2MV}{m+M})
This means that if the bat is ligther than the ball, the faster you throw, the slower the ball returns, or it doesn’t return at all, if 2MV is lower than (m-M)v. On the other hand, if the bat is heavier than the ball, the faster the ball moves initially, the faster it returns.
Of course, one shouldn’t disregard the batter completely. A better model is to assume that part of the batter is co-moving with the bat. Then, M is the aggregate mass of the bat and the movable part of the batter, which is very likely to exceed the mass of the ball, and then, the faster the ball goes, the faster it returns.
Even better model expects that part of the energy is lost on the bat-batter boundary and in the batter’s muscles and joints. Now we should have a reasonable model of how much this is. My hunch is that this amount in fact increases (absolutely, not only relatively) with increasing speed of the ball: if the ball is really fast, it will knock the bat out of the batter’s hands before the impulse could be transmitted between the batter’s body and the bat, making it effectively a free bat and ball system. But I don’t have any idea about actual numbers.
By the way, I envy you Americans such a Newtonian sport. It’s not so easy to modify those examples to naturally fit soccer or ice hockey.
Edit: I have now found that the bats are a lot heavier than the balls, which makes all speculations about the batter’s physiology irrelevant to the question.
Intuition pump: if the masses are equal and one isn’t moving, then as per a Newtonian cradle, the moving ball stops completely and the other takes up its motion.
Well, if we ignore the force being exerted on the bat by the batter, then we can use the principle of conservation of momentum, and if we make-believe that the collision is perfectly elastic, then we can say that kinetic energy is conserved as well. This gives us two equations to play with:
m_{bat}v_{bat} m_{ball}v_{ball}=m_{bat}w_{bat} m_{ball}w_{ball}
frac{1}{2}m_{bat}v_{bat}2 frac{1}{2}m_{ball}v_{ball}2=frac{1}{2}m_{bat}w_{bat}2 frac{1}{2}m_{ball}w_{ball}2
where the vs are initial velocities, ms are masses, and ws are subsequent (after-collision) velocities. Algebra then lets us derive two equations for the subsequent velocity of the ball in terms of the masses of the bat and ball, the subsequent and initial velocities of the bat, and the initial velocity of the ball:
and
In both these equations [edit: this is wrong; see addendum below], the subsequent velocity of the ball is an increasing function of the initial velocity of the ball (when all the other variables are treated as constants). Clearly, balls that are hit faster will travel farther, so this would seem to suggest that the faster pitch will be hit farther. However, in addition to the simplifying assumptions we made at the beginning, this analysis also assumes that the after-collision velocity of the bat is independent of the speed of the pitch, which is surely not true, so this reasoning cannot be said to have definitively resolved the question.
EDITED TO ADD: Except velocity and momentum are vector quantities, so the initial velocity of the ball should be regarded as having sign opposite of that of the initial velocity of the bat. This would seem to mean that our momentum equation is actually saying that the slower pitch will be hit further … I’m confused.
No reason to be confused, remember you have ignored the force exerted by the batter. If the ball is enough heavy and fast, it will simply shoot away the bat and continue forward. (Edit: the mistake you made is
which you can’t, since the posterior bat velocity isn’t independent of the prior ball velocity.)
Let’s make it explicit:
Let w and W are velocities of the ball and the bat after the hit, v and V are the velocities before the hit, m and M are masses of the ball and the bat, respectively. Then we have
v+2MV}{m+M})This means that if the bat is ligther than the ball, the faster you throw, the slower the ball returns, or it doesn’t return at all, if 2MV is lower than (m-M)v. On the other hand, if the bat is heavier than the ball, the faster the ball moves initially, the faster it returns.
Of course, one shouldn’t disregard the batter completely. A better model is to assume that part of the batter is co-moving with the bat. Then, M is the aggregate mass of the bat and the movable part of the batter, which is very likely to exceed the mass of the ball, and then, the faster the ball goes, the faster it returns.
Even better model expects that part of the energy is lost on the bat-batter boundary and in the batter’s muscles and joints. Now we should have a reasonable model of how much this is. My hunch is that this amount in fact increases (absolutely, not only relatively) with increasing speed of the ball: if the ball is really fast, it will knock the bat out of the batter’s hands before the impulse could be transmitted between the batter’s body and the bat, making it effectively a free bat and ball system. But I don’t have any idea about actual numbers.
By the way, I envy you Americans such a Newtonian sport. It’s not so easy to modify those examples to naturally fit soccer or ice hockey.
Edit: I have now found that the bats are a lot heavier than the balls, which makes all speculations about the batter’s physiology irrelevant to the question.
Intuition pump: if the masses are equal and one isn’t moving, then as per a Newtonian cradle, the moving ball stops completely and the other takes up its motion.
Also, hooray for Maxima!