(18a) Newton's 3rd LawThis lesson presents and explains Newton's 3rd law ("To every action, an equal and opposite reaction"), gives some examples as well as an (optional) formulation of Newton's laws that avoid explicitly mentioning forces. The next lesson (#18b) applies one of the most useful consequences of the 3rd law, namely the conservation of momentum. 
Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern
This lesson plan supplements: section #18a "Newton's 3rd law" http://www.phy6.org/stargaze/Snewton3.htm
Home pages: 
Goals: The student will learn
The formal definition of Newton's 3rd law: "forces always originate in pairs, equal in magnitude and opposite in direction."
Rotation of a sprinkler  The reason it may take 3 strong fire fighters to hold and aim a big hose (it kicks back!) The reason it is unwise to jump from boat to shore before tying the boat up (the reaction to the force of your leg muscles pushing you forward will push the boat away from shore!). Why you cannot balance a stopped bike by shifting your weight. Optional: how one can formally avoid the notion of force in formulating Newton's 2nd and 3rd laws. Starting the lesson Start by making clear the different nature and application of Newton's 2nd and 3rd laws (the students might copy from the blackboard some or all the words below): Newton's 2nd law describes what happens when a force is applied to an object, but says nothing about the way such a force originates. Newton's 3rd law discusses the origin of forces: they are always created in pairs, equal in magnitude and opposite in direction. It always involves more than one object. Terms: Newton's 3rd law, recoil.
Guiding questions and additional tidbits:
What does Newton's 3rd law state?
The student should explain it is something like "force" (which in its turn may be described as "the cause of motion, expressed quantitatively")
Wouldn't a pair of equal and opposite forces cancel each other?
It states that if one objectlet's call it "A"pushes another onesay "B"with force F, then "B" pushes "A" back with force –F. The two forces have the same strength, but the negative sign shows one has the opposite direction of the other. (Do not go past here until everyone in class understands this idea!) We will go into the mathematics later, but first, some examples which make the concept clear. Can anyone give an example? Examples should involve forces that actually cause motion.
 A rising rocket. One force is the pressure pushing a fast jet of burning gas out of the back of the rocket. This leads to an equal but opposing force, the thrust on the rocket itself.
[Around 1900 a popular notion arose that rockets could not travel in empty space "because they lacked air to push against." Dr. Robert Goddard finally demonstrated experimentally that this was false. Actually, a rocket flies better in empty space, since it has no air resistance to overcome.]
 The rotation of a garden sprinklerthe kind which has 23 arms pivoted at the middle, with each arm bending near its end in the direction opposite to that of the rotation. Draw on the blackboard. How does this work?
They never managed to do it, or to agree on predicting the result. Later others tried itthe process is complicated by other factors and the rotation is either absent or very slow.]
A big fire hose always has long handles on both sides of the nozzle. Why?
When jumping from a boat to shore, it is always advisable to tie up the boat before jumping. Why?
You sit on a bike which is not moving, and it starts to fall towards the left. Can you balance it leaning your body to the right?
Riders on a moving bike balance it by turning the front wheel left or right. Because of conservation of angular momentum, such motions rotate the entire bike and its rider, around the line on which the wheels touch the ground, and such rotation can straightenup the bike.]
(End of examples) Newton's 3rd law speaks of "equal and opposite forces." The meaning of "opposite directions" is clearbut how can one show the forces are equally strong? For this you need Newton's 2nd law, and here is how.  But first, a few words about notation. Both Newton's 2nd and 3rd laws are vector equations, involving forces and accelerations, vector quantities which have directions as well as magnitude. The same letter in ordinary font may be used for the magnitude of that vector.
Books often use bold face letters to distinguish vectors. On the blackboard we use instead a wavy line above the letter (some teachers use an underline instead; either is OK).
Let us call the force F, with magnitude F, and suppose we have two objects (e.g. billiard balls) labeled "A" and "B", pushing each other apart.
Suppose "A" has mass M_{1} and undergoes acceleration a_{1}
Let us first look just at magnitudeforget the vector character, forget the minus. Then.. .. Without the minus sign the same F appears on the left! Therefore
Divide both equations by M_{2} .. .. .. and then divide both by a_{1} .. .. .. You can see the result: when only "A" and "B" are involved, their accelerations always have the same ratio. Say, M_{1} is a heavy billiard ball collide with a light one of mass M_{2}. If the collision is a fast one, they fly off rapidly, if a gentle one, they fly off slowly, but in each case the ratio between the accelerations is the same!
 Which ball one has the greater acceleration?
Now let's repeat with the minus sign: we have [Optional]
In this formulation, the two laws are replaced by the following:
That ratio defines the ratio of the masses, through the equation derived earlier
If M_{2} is a liter of water (1000 cubic centimetersdefinition of the kilogram), then the above equation provides a way of defining the mass M_{1} of the other object, "through the back door." With mass defined, we can also define the unit of forcethe Newtonas that which causes a mass of 1 kilogram an acceleration of 1 meter/second2. Thus mass and force can be defined as convenient secondary quantities, but the fundamental law only involves measurable accelerations.

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Author and Curator: Dr. David P. Stern
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