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I have been doing personal study in Classical Mechanics and reading Newton's Laws. While thinking about them I had a question I haven't been able to answer. It comes from the interactions of Newton's 1st and 3rd laws.

First law

In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

Third law

When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

So here's the problem: An object is in a certain state (at rest or in motion with zero acceleration) and an external force is applied to that object. By the 1st Law the object should experience a change of state. But then by the 3rd Law a reactive force, of equal magnitude and opposite direction, would be applied on the object and now the sum of the forces would be zero (the object's state doesn't change at all). By this reasoning it would be impossible for an object to experience a state change because these forces would be in constant opposition.

But objects do experience state changes so where is the mistake in this reasoning? Thank you for any help!

Gert
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Jassaway
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  • You said it yourself. "When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body." – pho Sep 01 '20 at 21:46

3 Answers3

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This is a classic misunderstanding of the third law. The mathematical statement of the law is that if you have two objects, then $$\mathbf{F}_{12} = - \mathbf{F}_{21},$$ in other words, the force of 1 on 2 is equal and opposite to the force of 2 on 1.

The two forces do not act on the same object, but on different objects. In your example, you assume the reaction force to act on the same object, but this is incorrect. Indeed, if this were true (as you point out), nothing would ever move!

An example of the "third law" would be, for example, that the force that the Sun exerts on the Earth is the same as the force that the Earth exerts on the sun, but in the opposite direction.

EDIT: After writing my answer, I discovered a fantastic SE post here that explains it much better than I could -- Given Newton's third law, why are things capable of moving?

Philip
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Indeed as @Philip pointed out it is a classic misunderstanding of the third law. We keep seeing this over and over again on this site. Clearly, the teaching of the third law at the high school and/or undergraduate level appears to be inadequate based on the number of times this comes up.

To expand on @Philip equation, it all becomes clear if you combine the third law with the second law. From the third law

$$F_{12}=F_{21}=F$$

If the mass of object 1 is $M_1$ and the mass of object 2 is $M_2$, and there are no other forces acting upon each object other than the force of the other object, then from Newton's second law:

$$a_{1}=\frac{F}{M_1}$$

$$a_{2}=\frac{F}{M_2}$$

If $$M_{1}>M_2$$

Then

$$a_{2}>a_{1}$$

Hope this helps.

Bob D
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  • Thank you for the help. I see the mistake now. – Jassaway Sep 02 '20 at 02:42
  • "We keep seeing this over and over again on this site." Then it should have been easy to find a duplicate ;) I guess we can add this one to the list as well. – BioPhysicist Sep 02 '20 at 03:03
  • @BioPhysicist Ouch! So you've never answered a duplicate? Even if it is a duplicate, you know sometimes a different explanation than previously given can click when others haven't. If previous answers on Newton 3 always did the job, maybe it wouldn't keep coming up. – Bob D Sep 03 '20 at 12:33
  • I was just messing around. The odd thing here is that the OP directly quotes the third law that says which forces act on which bodies. It might be more about reading comprehension at this point. – BioPhysicist Sep 03 '20 at 13:35
  • @BioPhysicist OK, I understand. – Bob D Sep 03 '20 at 13:37
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A simple example would be if you have two balls rolling towards each other. When they collide they change velocity and bounce off in different directions.

1st Law: Velocity of ball A changes thanks to ball B, acting as an outside force, colliding with ball A.

The 3rd law: ball B also gets a change in velocity "equal and opposite" to that of ball A's.

The 3rd law doesn't apply back to the first object, it states that the first object imparts an equal and opposite force onto the object that made it change its velocity.

Corey
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