Newton's Third Law of Motion: Difference between revisions

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m Fixed a major misunderstanding regarding internal vs. external forces in the system diagrams and wrote a much more concise, correct description of how they cancel out. Completely rebuilt the math sections using specific backend tags to fix the broken display and make sure every formula looks clean and sharp. Hand-aligned all the multi-step physics problems so the math actually lines up vertically, making the solutions way easier to follow. Corrected the confusing explanation of Newton's Third
Line 1: Line 1:
By Karan Shah
By Alexander Wasil
 


==Main Idea==
==Main Idea==


Newton’s Third Law of Motion describes a push or pull that acts on an object as a result of its interaction with another object. According to this law, for every action there is an equal and opposite re-action. This means that for every force there is a reaction force that is equal in size, but opposite in direction. Meaning that when an object 1 pushes another object 2, object 1 gets pushed back with equal force but in the opposite direction.  
Newton’s Third Law of Motion states that all forces occur in pairs as a result of an interaction between two objects. When object A exerts a force on object B, object B simultaneously exerts a force equal in magnitude and opposite in direction on object A. These forces are part of a single interaction; neither exists without the other.


The third law of motion is also referred to as the action-reaction law because both objects are part of a single interaction and neither force can exist without the other. An important concept to remember about Newton's Third Law of Motion is that the two forces are of the same type. For example, when you throw a ball in the sky the Earth exerts a gravitational force on the ball and the ball also exerts a gravitational force that is equal in magnitude and opposite in direction on the earth. Another example that can sum up the concept of Newton's Third Law is when you walk. When you push down upon the ground and ground pushes with the same force upward. Similarly, the tires of a car push against the road while the road pushes back on the tires.
A critical component of this law is that the two forces are always of the exact same type. If object A exerts a normal force on object B, object B exerts a normal force on object A. Furthermore, these force pairs never act on the same object. This distinction is necessary when drawing Free Body Diagrams, as internal forces within a defined system cancel each other out.
===A Mathematical Model===


A simple equation can be used to model Newton's Third Law of Motion.
Common examples of this interaction include gravitational pull and contact forces. The Earth exerts a downward gravitational force on a projectile, and the projectile exerts an equal upward gravitational force on the Earth. When a person walks, they exert a backward frictional force on the ground, and the ground exerts an equal forward frictional force on the person.


<math> F_{1 on 2} = -F_{2 on 1} </math>


A force is reciprocated with an equal and opposite force.
For example, someone that weighs 130 pounds puts a force of 130 pounds onto the ground, and the normal force of the ground on the person is also 130 pounds.


<math> 130 = 130 </math>
===A Mathematical Model===


===A Computational Model===
The law is modeled using vector notation to account for both magnitude and direction.


https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html
<math>\vec{F}_{A \text{ on } B} = -\vec{F}_{B \text{ on } A}</math>


This model shows how a collision models Newton's Third Law. The force of the first ball on the second ball is equal to the force of the second ball on the first ball. The difference in velocity and acceleration is a result of the difference in mass between the two balls.
Because the forces are equal and opposite, their sum is zero when considering the two objects as a single system.


==Examples==
As an example using SI units, consider a person with a mass of 60 kg standing on a flat surface. The gravitational force (weight) acting on the person is approximately 588 N downward. The person exerts a 588 N contact force downward onto the ground. Simultaneously, the ground exerts a normal force of 588 N upward onto the person.


Here are some problems regarding Newton's Third Law.
<math>| -588 \text{ N} | = | 588 \text{ N} |</math>


===Simple===
===A Computational Model===


'''Question'''
[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html PhET Collision Lab]


[[File: Snip20151128_10.png| thumb | left | 250px |Simple Example]]Car B is stopped at a red light. The brakes in Car A have failed and Car A is coming towards Car B at 60 kmh. Car A then runs into the back of Car B, What can be said about the force on Car A on Car B and the force on Car B on Car A?
This interactive simulation demonstrates the Third Law during elastic and inelastic collisions. To observe the law in action, set the masses of the two spheres to different values and enable the force vector arrows. During the impact, the force arrows will remain equal in length and opposite in direction regardless of the mass disparity. The difference in the resulting velocities and accelerations is due to the difference in mass, as described by Newton's Second Law, not a difference in the applied forces.


[[File:Suggested_Glowscript_Collision_Model.png|thumb|right|300px|Placeholder: A GlowScript model showing two colliding spheres with continuously updating force arrows.]]


==Examples==


===Simple===


'''Question'''


[[File: Snip20151128_10.png| thumb | left | 250px |Car collision diagram showing Car A approaching Car B]]
Car B is stopped at a red light. The brakes in Car A have failed, and Car A is traveling toward Car B at 60 km/h. Car A collides with the back of Car B. What is the relationship between the force Car A exerts on Car B and the force Car B exerts on Car A?


'''Answer'''
'''Answer'''


B exerts the same amount of force on A as A exerts on B.  
Car B exerts the exact same amount of force on Car A as Car A exerts on Car B. The forces are equal in magnitude but act in strictly opposite directions.
Just the direction of the forces will be in the opposite direction.


===Middling===
===Middling===


'''Question'''


'''Question'''
Blocks with masses of 1.0 kg, 2.0 kg, and 3.0 kg are lined up in a row on a frictionless horizontal table. All three are pushed forward by an 8.0 N applied force pushing on the 1.0 kg block.
<br>(a) How much force does the 2.0 kg block exert on the 3.0 kg block?
<br>(b) How much force does the 2.0 kg block exert on the 1.0 kg block?


Blocks with masses of 1 kg, 2 kg, and 3 kg are lined up in a row on a frictionless table. All three are pushed forward by a 8 N force applied to the 1 kg block. (a) How much force does the 2 kg block exert on the 3 kg block? (b) How much force does the 2 kg block exert on the 1 kg block?




'''Answer'''
'''Answer'''


(a)  
(a)
Find the Acceleration of the Whole Object:
<br>First, define the system as all three blocks to find the total acceleration.
Total Mass: 6kg
<br>Total Mass: <math>1.0 \text{ kg} + 2.0 \text{ kg} + 3.0 \text{ kg} = 6.0 \text{ kg}</math>
8 = (6) a
a = 8 / 6 = 1.33 m/s^2
Total Acceleration: 1.33 m/s^2 (Acceleration will be the same for all three blocks)
F(2 on 3) = m(3) * a 
3 * 1.33 = 3.999 N
 
(b)
Total Acceleration: 1.33 m/s^2
Mass to push: 5 kg (Because we are also pushing the 3 kg block)
F(1 on 2) = 5 * 1.33
F(1 on 2) = 5.33 N


===Difficult===
<math>
\begin{aligned}
F_{\text{net}} &= m_{\text{total}} \cdot a \\
8.0 \text{ N} &= (6.0 \text{ kg}) \cdot a \\
a &= 1.33 \text{ m/s}^2
\end{aligned}
</math>


'''Question'''
The acceleration is <math>1.33 \text{ m/s}^2</math> for all blocks in the system.
<br>To find the force of block 2 on block 3, define block 3 as the system.


A massive steel cable drags a 30 kg block across a horizontal, frictionless surface. A 100 N force applied to the cable causes the block to reach a speed of 5.0 m/s in a distance of 5.0 m. What is the mass of the cable?
<math>
\begin{aligned}
F_{2 \text{ on } 3} &= m_3 \cdot a \\
F_{2 \text{ on } 3} &= (3.0 \text{ kg}) \cdot (1.33 \text{ m/s}^2) \\
F_{2 \text{ on } 3} &= 4.0 \text{ N}
\end{aligned}
</math>


'''Answer'''
(b)
<br>To find the force of block 1 on block 2, define blocks 2 and 3 as the combined system being pushed by block 1.
<br>System mass: <math>2.0 \text{ kg} + 3.0 \text{ kg} = 5.0 \text{ kg}</math>


acceleration of the cable and block: a = V² / (2x) = (5²) / (2)(5) = 2.5 m/s²
<math>
solve for mass: F = M * a → 100 = (30+m) * 2.5
\begin{aligned}
                m = 10 kg
F_{1 \text{ on } 2} &= (5.0 \text{ kg}) \cdot (1.33 \text{ m/s}^2) \\
F_{1 \text{ on } 2} &= 6.65 \text{ N}
\end{aligned}
</math>


==Connectedness==
According to Newton's Third Law, the force of block 2 on block 1 is equal and opposite to the force of block 1 on block 2.


<math>|F_{2 \text{ on } 1}| = |F_{1 \text{ on } 2}| = 6.7 \text{ N}</math>


Newton's Third Law is connected to the concept of a spacecraft flying in space. When a spacecraft fires a thruster rocket, the exhaust gas pushes against the thruster and the thruster pushes against the exhaust gas. The gas and rocket move in opposite directions. This is an example of Newton's Third Law because both forces are equal in magnitude and opposite in direction. While most systems that utilize the change in velocity of one object in the system to change the velocity of the other object. Rockets, on the other hand, utilize the change in mass of the fuel due to burning to increase the speed on the rocket, to achieve the high velocity necessary to escape into the Earth's atmosphere. This is modeled by the formula <math> mv = mv </math>. A decrease in mass directly causes an increase in velocity.
===Difficult===


==History==
'''Question'''


A massive steel cable drags a 30.0 kg block across a horizontal, frictionless surface. A 100.0 N force applied to the cable causes the block to reach a speed of 5.0 m/s over a distance of 5.0 m. What is the mass of the cable?


[[File:Sir Isaac Newton (1643-1727).jpg|right|200px|Sir Isaac Newton (1643-1727)]] Sir Isaac Newton was a renowned scientist and mathematician who helped create a foundation for modern studies. He was born in England in 1643 and worked his way to earn a bachelor’s and master’s degree from Trinity College Cambridge. He was highly interested in math, physics, and astronomy and wrote many of his ideas in a journal. One of those ideas was about the three laws of motion. In 1687 Isaac Newton made his work on his book, Philosophiae Naturalis Principia Mathematica or Principia, known to the public. He discussed the principles of time, force, and motion that helped create modern physical science and helped account for much of the phenomena viewed in the world. Some of the principles he discusses include acceleration, initial movement, fluid dynamics, and motion. Newton’s Laws first appeared in the Principia and discussed the relationship that exists between forces acting on a body and the motion of the body. For the third law, he stated that for every action/force in nature, there will be an equal and opposite reaction.
'''Answer'''


== See also ==
Calculate the acceleration of the cable and block system using kinematics:


===Further reading===
<math>
\begin{aligned}
v^2 &= v_0^2 + 2a\Delta x \\
(5.0 \text{ m/s})^2 &= 0 + 2(a)(5.0 \text{ m}) \\
a &= 2.5 \text{ m/s}^2
\end{aligned}
</math>


Books, Articles or other print media on this topic
Apply Newton's Second Law to the combined system to solve for the cable's mass (<math>m</math>):


[http://www.wired.com/2013/10/a-closer-look-at-newtons-third-law/ A Closer Look at Newton’s Third Law]
<math>
\begin{aligned}
F_{\text{net}} &= m_{\text{total}} \cdot a \\
100.0 \text{ N} &= (30.0 \text{ kg} + m) \cdot (2.5 \text{ m/s}^2) \\
40.0 &= 30.0 + m \\
m &= 10.0 \text{ kg}
\end{aligned}
</math>


[http://phys.org/news/2015-05-newton-law-broken.html What happens when Newton's third law is broken?]
==Connectedness==


[http://www.livestrong.com/article/423739-newtons-three-laws-motion-used-baseball/ How Are Newton's Three Laws of Motion Used in Baseball?]
Newton's Third Law is foundational to the Conservation of Momentum. Because interacting objects exert equal and opposite forces on each other (<math>\vec{F}_{1} = -\vec{F}_{2}</math>) for the exact same duration (<math>\Delta t</math>), their respective changes in momentum are equal and opposite (<math>\Delta\vec{p}_1 = -\Delta\vec{p}_2</math>). In an isolated system, the total momentum remains constant.


[https://www.newscientist.com/article/dn24411-light-can-break-newtons-third-law-by-cheating/ Light can break Newton’s third law – by cheating]
This principle applies directly to spacecraft propulsion. When a spacecraft fires a thruster, the engine exerts a force on the exhaust gas, and the exhaust gas exerts an equal and opposite force on the engine. Most terrestrial vehicles rely on interacting with an external surface, such as a road, to change velocity. Spacecraft operate in a vacuum and instead rely on ejecting mass. This relationship is modeled by the thrust equation:


[http://science360.gov/obj/video/d0e16d27-05d4-4511-9394-2758aa066981/science-nfl-football-newtons-third-law-motion Science of Football]
<math>\vec{F}_{\text{thrust}} = -\vec{v}_{\text{exhaust}} \frac{dm}{dt}</math>


===External links===


Internet resources on this topic


[http://teachertech.rice.edu/Participants/louviere/Newton/law3.html The Third Law of Motion]
A decrease in the mass of the system due to expelled fuel generates the force required to increase the velocity of the rocket.


[http://www.physicsclassroom.com/class/newtlaws/Lesson-4/Newton-s-Third-Law Newton's Third Law of Motion]
==History==


[https://www.grc.nasa.gov/www/k-12/WindTunnel/Activities/third_law_motion.html Newton's Third Law of Motion]
[[File:Principia_Title_Page_Suggested.jpg|right|200px|Title page of the first edition of Philosophiæ Naturalis Principia Mathematica]]
The Third Law of Motion was formalized by Sir Isaac Newton in his seminal 1687 work, ''Philosophiæ Naturalis Principia Mathematica'' (commonly known as the ''Principia''). Prior to this publication, the relationship between interacting forces and planetary motion lacked a unified mathematical framework. In the ''Principia'', Newton defined the principles of time, force, and motion, laying the groundwork for classical mechanics.


==References==
The Third Law specifically resolved the issue of how forces operate as interactions between bodies rather than isolated properties of single objects. By stating that every action has an equal and opposite reaction, Newton provided the necessary logic to explain universal gravitation and the conservation of momentum, allowing for accurate predictions of both terrestrial phenomena and celestial orbits.


This section contains the the references you used while writing this page
== See also ==


Knight, R., & Jones, B. (n.d.). College physics: A strategic approach (Third edition, Global ed.).
===Further reading===


http://www.physicsclassroom.com/class/newtlaws/Lesson-4/Newton-s-Third-Law
* [http://www.wired.com/2013/10/a-closer-look-at-newtons-third-law/ A Closer Look at Newton’s Third Law]
* [http://phys.org/news/2015-05-newton-law-broken.html What happens when Newton's third law is broken?]
* [http://www.livestrong.com/article/423739-newtons-three-laws-motion-used-baseball/ How Are Newton's Three Laws of Motion Used in Baseball?]
* [https://www.newscientist.com/article/dn24411-light-can-break-newtons-third-law-by-cheating/ Light can break Newton’s third law – by cheating]
* [http://science360.gov/obj/video/d0e16d27-05d4-4511-9394-2758aa066981/science-nfl-football-newtons-third-law-motion Science of Football]


https://www.grc.nasa.gov/www/k-12/airplane/newton3.html
===External links===


http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html
* [http://teachertech.rice.edu/Participants/louviere/Newton/law3.html The Third Law of Motion]
* [http://www.physicsclassroom.com/class/newtlaws/Lesson-4/Newton-s-Third-Law Newton's Third Law of Motion]
* [https://www.grc.nasa.gov/www/k-12/WindTunnel/Activities/third_law_motion.html Newton's Third Law of Motion]


http://science360.gov/obj/video/d0e16d27-05d4-4511-9394-2758aa066981/science-nfl-football-newtons-third-law-motion
==References==


http://www.livescience.com/46561-newton-third-law.html
* Knight, R., & Jones, B. (n.d.). College physics: A strategic approach (Third edition, Global ed.).
* http://www.physicsclassroom.com/class/newtlaws/Lesson-4/Newton-s-Third-Law
* https://www.grc.nasa.gov/www/k-12/airplane/newton3.html
* http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html
* http://science360.gov/obj/video/d0e16d27-05d4-4511-9394-2758aa066981/science-nfl-football-newtons-third-law-motion
* http://www.livescience.com/46561-newton-third-law.html


[[Category:Collisions]]
[[Category:Collisions]]

Revision as of 22:10, 28 April 2026

By Alexander Wasil

Main Idea

Newton’s Third Law of Motion states that all forces occur in pairs as a result of an interaction between two objects. When object A exerts a force on object B, object B simultaneously exerts a force equal in magnitude and opposite in direction on object A. These forces are part of a single interaction; neither exists without the other.

A critical component of this law is that the two forces are always of the exact same type. If object A exerts a normal force on object B, object B exerts a normal force on object A. Furthermore, these force pairs never act on the same object. This distinction is necessary when drawing Free Body Diagrams, as internal forces within a defined system cancel each other out.

Common examples of this interaction include gravitational pull and contact forces. The Earth exerts a downward gravitational force on a projectile, and the projectile exerts an equal upward gravitational force on the Earth. When a person walks, they exert a backward frictional force on the ground, and the ground exerts an equal forward frictional force on the person.


A Mathematical Model

The law is modeled using vector notation to account for both magnitude and direction.

[math]\displaystyle{ \vec{F}_{A \text{ on } B} = -\vec{F}_{B \text{ on } A} }[/math]

Because the forces are equal and opposite, their sum is zero when considering the two objects as a single system.

As an example using SI units, consider a person with a mass of 60 kg standing on a flat surface. The gravitational force (weight) acting on the person is approximately 588 N downward. The person exerts a 588 N contact force downward onto the ground. Simultaneously, the ground exerts a normal force of 588 N upward onto the person.

[math]\displaystyle{ | -588 \text{ N} | = | 588 \text{ N} | }[/math]

A Computational Model

PhET Collision Lab

This interactive simulation demonstrates the Third Law during elastic and inelastic collisions. To observe the law in action, set the masses of the two spheres to different values and enable the force vector arrows. During the impact, the force arrows will remain equal in length and opposite in direction regardless of the mass disparity. The difference in the resulting velocities and accelerations is due to the difference in mass, as described by Newton's Second Law, not a difference in the applied forces.

File:Suggested Glowscript Collision Model.png
Placeholder: A GlowScript model showing two colliding spheres with continuously updating force arrows.

Examples

Simple

Question

Car collision diagram showing Car A approaching Car B

Car B is stopped at a red light. The brakes in Car A have failed, and Car A is traveling toward Car B at 60 km/h. Car A collides with the back of Car B. What is the relationship between the force Car A exerts on Car B and the force Car B exerts on Car A?

Answer

Car B exerts the exact same amount of force on Car A as Car A exerts on Car B. The forces are equal in magnitude but act in strictly opposite directions.

Middling

Question

Blocks with masses of 1.0 kg, 2.0 kg, and 3.0 kg are lined up in a row on a frictionless horizontal table. All three are pushed forward by an 8.0 N applied force pushing on the 1.0 kg block.
(a) How much force does the 2.0 kg block exert on the 3.0 kg block?
(b) How much force does the 2.0 kg block exert on the 1.0 kg block?


Answer

(a)
First, define the system as all three blocks to find the total acceleration.
Total Mass: [math]\displaystyle{ 1.0 \text{ kg} + 2.0 \text{ kg} + 3.0 \text{ kg} = 6.0 \text{ kg} }[/math]

[math]\displaystyle{ \begin{aligned} F_{\text{net}} &= m_{\text{total}} \cdot a \\ 8.0 \text{ N} &= (6.0 \text{ kg}) \cdot a \\ a &= 1.33 \text{ m/s}^2 \end{aligned} }[/math]

The acceleration is [math]\displaystyle{ 1.33 \text{ m/s}^2 }[/math] for all blocks in the system.
To find the force of block 2 on block 3, define block 3 as the system.

[math]\displaystyle{ \begin{aligned} F_{2 \text{ on } 3} &= m_3 \cdot a \\ F_{2 \text{ on } 3} &= (3.0 \text{ kg}) \cdot (1.33 \text{ m/s}^2) \\ F_{2 \text{ on } 3} &= 4.0 \text{ N} \end{aligned} }[/math]

(b)
To find the force of block 1 on block 2, define blocks 2 and 3 as the combined system being pushed by block 1.
System mass: [math]\displaystyle{ 2.0 \text{ kg} + 3.0 \text{ kg} = 5.0 \text{ kg} }[/math]

[math]\displaystyle{ \begin{aligned} F_{1 \text{ on } 2} &= (5.0 \text{ kg}) \cdot (1.33 \text{ m/s}^2) \\ F_{1 \text{ on } 2} &= 6.65 \text{ N} \end{aligned} }[/math]

According to Newton's Third Law, the force of block 2 on block 1 is equal and opposite to the force of block 1 on block 2.

[math]\displaystyle{ |F_{2 \text{ on } 1}| = |F_{1 \text{ on } 2}| = 6.7 \text{ N} }[/math]

Difficult

Question

A massive steel cable drags a 30.0 kg block across a horizontal, frictionless surface. A 100.0 N force applied to the cable causes the block to reach a speed of 5.0 m/s over a distance of 5.0 m. What is the mass of the cable?

Answer

Calculate the acceleration of the cable and block system using kinematics:

[math]\displaystyle{ \begin{aligned} v^2 &= v_0^2 + 2a\Delta x \\ (5.0 \text{ m/s})^2 &= 0 + 2(a)(5.0 \text{ m}) \\ a &= 2.5 \text{ m/s}^2 \end{aligned} }[/math]

Apply Newton's Second Law to the combined system to solve for the cable's mass ([math]\displaystyle{ m }[/math]):

[math]\displaystyle{ \begin{aligned} F_{\text{net}} &= m_{\text{total}} \cdot a \\ 100.0 \text{ N} &= (30.0 \text{ kg} + m) \cdot (2.5 \text{ m/s}^2) \\ 40.0 &= 30.0 + m \\ m &= 10.0 \text{ kg} \end{aligned} }[/math]

Connectedness

Newton's Third Law is foundational to the Conservation of Momentum. Because interacting objects exert equal and opposite forces on each other ([math]\displaystyle{ \vec{F}_{1} = -\vec{F}_{2} }[/math]) for the exact same duration ([math]\displaystyle{ \Delta t }[/math]), their respective changes in momentum are equal and opposite ([math]\displaystyle{ \Delta\vec{p}_1 = -\Delta\vec{p}_2 }[/math]). In an isolated system, the total momentum remains constant.

This principle applies directly to spacecraft propulsion. When a spacecraft fires a thruster, the engine exerts a force on the exhaust gas, and the exhaust gas exerts an equal and opposite force on the engine. Most terrestrial vehicles rely on interacting with an external surface, such as a road, to change velocity. Spacecraft operate in a vacuum and instead rely on ejecting mass. This relationship is modeled by the thrust equation:

[math]\displaystyle{ \vec{F}_{\text{thrust}} = -\vec{v}_{\text{exhaust}} \frac{dm}{dt} }[/math]


A decrease in the mass of the system due to expelled fuel generates the force required to increase the velocity of the rocket.

History

Title page of the first edition of Philosophiæ Naturalis Principia Mathematica
Title page of the first edition of Philosophiæ Naturalis Principia Mathematica

The Third Law of Motion was formalized by Sir Isaac Newton in his seminal 1687 work, Philosophiæ Naturalis Principia Mathematica (commonly known as the Principia). Prior to this publication, the relationship between interacting forces and planetary motion lacked a unified mathematical framework. In the Principia, Newton defined the principles of time, force, and motion, laying the groundwork for classical mechanics.

The Third Law specifically resolved the issue of how forces operate as interactions between bodies rather than isolated properties of single objects. By stating that every action has an equal and opposite reaction, Newton provided the necessary logic to explain universal gravitation and the conservation of momentum, allowing for accurate predictions of both terrestrial phenomena and celestial orbits.

See also

Further reading

External links

References

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