How Does Newton's Third Law Apply When a Bat Hits a Ball?

[SillyYak]

The following reasoning leads to an apparent paradox; explain what’s wrong with the logic. A baseball player hits a ball. The ball and the bat spend a fraction of a second in contact. During that time they’re moving together, so their accelerations must be equal. Newton’s third law says that their forces on each other are also equal. But a = F/m, so how can this be, since their masses are unequal? (Note that the paradox isn’t resolved by considering the force of the batter’s hands on the bat. Not only is this force very small compared to the ball-bat force, but the batter could have just thrown the bat at the ball.)

I don't understand how to answer this question, if the bat and ball are changing their velocities at the same rate and the forces are the same don't masses have to be the same?

 

[DrClaude]

I don't understand how to answer this question, if the bat and ball are changing their velocities at the same rate and the forces are the same don't masses have to be the same?

The masses not being the same is the one thing that can be measured independently. Therefore, one of the other assumptions must be wrong.

 

[Janus]

I'll say this: It is one of those apparent paradoxes that arises when you try to treat real world objects as if they were ideal ones.

 

[haruspex]

During that time they’re moving together,

If you treat the two objects as perfectly rigid then the accelerations they undergo are infinite and last for zero time. The ball instantly gains velocity (in the bat's direction) while the bat instantly loses it.
If you insist on a duration for the accelerations then you must treat one or both objects as deformable. That allows their mass centres to have opposite accelerations while the points of contact have the same velocity.

 

[SillyYak]

The masses not being the same is the one thing that can be measured independently. Therefore, one of the other assumptions must be wrong.

well i was thinking since we know the force is the same and that the masses are definitely different the accelerations must not be the same, i have trouble visualizing this though

 

[SillyYak]

If you treat the two objects as perfectly rigid then the accelerations they undergo are infinite and last for zero time. The ball instantly gains velocity (in the bat's direction) while the bat instantly loses it.
If you insist on a duration for the accelerations then you must treat one or both objects as deformable. That allows their mass centres to have opposite accelerations while the points of contact have the same velocity.

hmm ok so from my understanding you are saying that if we take into account the amount each object deforms, the accelerations would turn out to be different after all?

 

[haruspex]

hmm ok so from my understanding you are saying that if we take into account the amount each object deforms, the accelerations would turn out to be different after all?

Yes, because their mass centres can have different accelerations from that of the common point of contact.

 

[SillyYak]

Yes, because their mass centres can have different accelerations from that of the common point of contact.

Thank you Guys i think i understand, so if the objects were let's say perfectly rigid would there be a violation of F=MA or would there be some other way of explaining the phenomenon

 

[Janus]

Thank you Guys i think i understand, so if the objects were let's say perfectly rigid would there be a violation of F=MA or would there be some other way of explaining the phenomenon

If they were perfectly rigid, then you could not say that they spent any measurable time (fraction of a second) in contact. All changes in velocities would be instantaneous. The ball would instantaneously change direction and the bat would instantaneously lose a bit of velocity on contact. But such changes in velocities in zero time should mean infinite acceleration for zero time for both bat and ball. The moment of contact is a discontinuity where the behaviors of the bat and ball are undefined.

 

[SillyYak]

If they were perfectly rigid, then you could not say that they spent any measurable time (fraction of a second) in contact. All changes in velocities would be instantaneous. The ball would instantaneously change direction and the bat would instantaneously lose a bit of velocity on contact. But such changes in velocities in zero time should mean infinite acceleration for zero time for both bat and ball. The moment of contact is a discontinuity where the behaviors of the bat and ball are undefined.

I see, Thank you Sir

 

[Merlin3189]

Can I just add one thing that no one seems to have commented on,

Newton’s third law says that their forces on each other are also equal.

I think he said, opposite. So whether their masses were the same or equal, their accelerations must have been opposite, not the same.

 

[Lukeblackhill]

I don't think the accelerations are the same. Although they'll be moving due to a same force, the difference of masses mean that the ball and the bat would face different accelerations. If we throw the bat with the ball when they collide, and no air resistance is measured in the surroundings, yet we would see the ball falling due to gravity further away than the bat would.

If you consider the mere fraction of a second in which they move together, you must consider them as the same body, where the question moves to a simple F (in both) = (sum of their masses) X (the common acceleration).