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

• SillyYak
In summary, there is an apparent paradox in the logic of a baseball player hitting a ball, as their masses are unequal but their accelerations must be equal according to Newton's third law. However, the paradox can be resolved by considering the deformability of the objects or treating them as perfectly rigid. This leads to different accelerations for the objects' mass centers and the point of contact. Additionally, it is important to note that the moment of contact is a discontinuity and the behaviors of the objects are undefined.
SillyYak
Poster warned about not using the homework template
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?

SillyYak said:
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.

SillyYak
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.

Merlin3189 and SillyYak
SillyYak said:
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.

DrClaude said:
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

haruspex said:
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?

SillyYak said:
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
haruspex said:
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

SillyYak said:
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
Janus said:
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

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.

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).

## 1. What is Newton's third law paradox?

Newton's third law states that for every action, there is an equal and opposite reaction. The paradox arises when we consider that if the forces are equal and opposite, how can there be any motion at all?

## 2. How does Newton's third law apply to everyday life?

Newton's third law is applicable to many everyday situations, such as walking, throwing a ball, or driving a car. For example, when you walk, you push the ground backwards with your foot, and the ground pushes you forward with an equal and opposite force, causing you to move forward.

## 3. Can Newton's third law be violated?

No, Newton's third law is a fundamental principle of physics and cannot be violated. It has been extensively tested and verified through numerous experiments and observations.

## 4. Are there any exceptions to Newton's third law?

While there are no exceptions to Newton's third law, there are situations where it may seem like one object is not experiencing an equal and opposite reaction. For example, when a rocket is launched, the force of the exhaust gases pushing the rocket forward may seem to be greater than the force of the rocket pushing back on the gases. However, this is because the rocket has a much larger mass than the gases, so the same force produces a smaller acceleration.

## 5. How does Newton's third law relate to conservation of momentum?

Newton's third law is closely related to the principle of conservation of momentum. Since every action has an equal and opposite reaction, the total momentum of a system remains constant. This means that when one object exerts a force on another, the other object will experience an equal and opposite force, resulting in a change in its momentum. This is why we see objects move in opposite directions when they collide.

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