Article 3. Implications of Newton’s Laws of Motion with Emphasis on the 2nd Law

Synopsis:  In this article we look briefly into the historical development of Newton’s laws of motion, with particular focus placed on Newton’s second law.   We show that equations expressing the Impulse=Momentum theorem and the Work=Energy theorem can both be directly derived from the second law.   In future articles we will apply these equations along with kinematic equations that describe fundamental relationships between time, distance, velocity, and acceleration to examine various aspects of the bow and arrow.

HISTORICAL BACKGROUND

One of the greatest advancements in the development of modern physics was made by Sir Isaac Newton (1642-1727) through the publishing of his book Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) in 1687.  In it, Newton put forth three laws which govern the motion of an object due to forces acting on it.  Although the original text was published in Latin, the first English translation was published shortly after Newton’s death by Englishman Andrew Motte in 1729 (Ref. 1).    Per Motte, Newton’s Three Laws of Motion are directly translated as follows:

• (1^st Law) Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

 

• (2^nd Law) The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

 

• (3^rd Law) To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts

In modern language, the first law describes the intrinsic property of matter called “inertia”.  Inertia resists the start of motion and also the change in magnitude and/or the direction of motion. Inertia is directly proportional to the mass of a body i.e., if the mass increases, the inertia also increases and it is also a function of the shape of the body.   We will discuss inertia in greater detail in future articles when dealing with the flight trajectory and penetration capacity of an arrow.

In regards to the third law, Newton himself describes it (again from Ref. 1) as:

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavor to relax or unbend itself, will draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinges upon another, and by its force change the motion of the other, that body also (became of the quality of, the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally proportional to the bodies.

Simply stated, the third law tells us that if a force is imparted on one object by another, there is an equal force imparted by the second object upon the first object in the opposite direction.

It is of interest to note that Newton did not develop his laws without the aid of previous researchers, in particular the Italian scientist Galileo Galilei and the German astronomer Johannes Kepler (Refs. 3, 4).  Newton is quoted as saying,

“if I have seen farther than others, it is because I was standing on the shoulders of giants," by way of thanking his predecessors for the contributions to science which made his Principia possible.

In fact, prior to Newton et al., it was generally accepted that the magnitude of the velocity of an object was directly proportional to the magnitude of an applied force, which was a concept first put forth by the Greek philosopher Aristotle (Ref. 2) nearly 2000 years prior to Newton.  In Aristotle’s concept, motion of an object occurred only as long as force was applied, and therefore motion stopped when the applied force was removed.  Newton on the other hand realized that an object can travel with constant momentum without a force being continuously applied.

IMPULSE=MOMENTUM THEOREM

Let’s proceed on to examining Newton’s second law where the change in momentum is due to an applied force.   Referring to the second law as stated above, it is important to note that when Newton uses the word motion, what he is describing (in modern terminology) is the physical property “momentum”, which is the product of mass and velocity; in particular, he is not universally using the term motion to describe merely the velocity of an object.    Also key in understanding Newton’s second law is that the description “alteration of motion” means that the momentum changes with time, and that the change in momentum is due to an applied force.   In modern scalar equation form where the direction of the momentum and the direction of the applied force are aligned, Newton’s second law can be stated as:

We can simplify Equation 1 by assuming that the mass is a constant and does not change with time, resulting in:

And, by multiplying through by , we obtain:

We can expand Equation 3 using discreet notation as:

Let’s examine Equation 4 closely.  On the left hand side of the equals sign, we have a force multiplied by an increment of time.  This is referred to as an Impulse.   On the right hand side, we have the momentum at the end of the time increment minus the momentum at the beginning of the time increment.  This relationship is called the Impulse=Momentum theorem and as we have shown, it is derived directly from Newton’s second law of motion.    Let’s carry Equation 4 one step further by rearranging it to get:

Notice what Equation 5 is telling us:  Initial momentum plus an impulse is equal to the final momentum. If the impulse occurs in the same direction as the initial momentum, then the final momentum will be greater than the initial momentum.  However, if the impulse occurs in the opposite direction of the initial momentum, (mathematically F will then have a negative value due to it being in the opposite direction of the initial momentum) then the final momentum will be less than the initial momentum.   A good example of this is when an arrow strikes a bone during the penetration event.   The arrow has an initial value of momentum before impact, the impact impulse (the force of the arrow impacting during the time interval delta t)  occurs to the arrow in the opposite direction of the initial momentum, and therefore the final momentum of the arrow is reduced after the bone has been penetrated.  

To summarize this section, we’ve shown via the progression of Equations 1-5 that beginning with a mathematical statement of Newton’s second law, we can derive an expression for the change in momentum of an object due to an applied force acting on the object over a defined interval of time.   

WORK=ENERGY THEOREM

We previously showed that a force applied to an object over a certain interval of time is equal to the change in momentum of that object.  Next, we will look at a similar relationship for when force is applied to an object over a certain interval of distance.  To derive this relationship algebraically, we will need the aid of kinematic equations (Ref. 5).   Kinematic equations describe fundamental physical relationships between time, distance, velocity, and acceleration.  It is common practice to mathematically manipulate the kinematic relationships to express for instance acceleration in terms of velocity and time, or just as easily, to express acceleration in terms of velocity and distance.    With the use of these kinematic relationships, the derivation proceeds as follows:

(Note: From here, the reader is welcome to follow along with the manipulation of these kinematic equations into a form that will plug directly into Newton’s second law.  If instead, you would prefer to skip to the end result, please scroll ahead to Equation 15, drb).

Starting as before with the expression for Newton’s second law:

we will be deriving a relationship for ∆𝑣/∆𝑡 in terms of velocity and distance. 

We know that the average velocity of an object can be determined by dividing the distance traveled by the time it takes to travel that distance.   In formula form, this is expressed as:

We can also define an average velocity given an initial velocity 𝑣𝑖 and a final velocity 𝑣𝑓:

In preparation for a later step (deriving Equation 11 from Equation 10 below), let’s set the right hand side of Equation 6 equal to the right hand side of Equation 7, and solve for 𝑡𝑓− 𝑡𝑖: 

Next, we need an expression for ∆𝑣/∆𝑡, since this term appears in Newton’s second law (Equation 2).  We start with the definition of ∆𝑣/∆𝑡:

Mathematically rearranging Equation 9 gives us:

We are now going to mathematically set Equations 8 equal to Equation 10, effectively eliminating the discrete time interval (𝑡𝑓− 𝑡𝑖 ):

Here, we are solving for ∆𝑣/∆𝑡, therefore:

After simplifying the left hand side of Equation 12, we arrive at the kinematic equation we’ve been looking for, which expresses the relationship between the change in velocity with respect to time in terms of the change in velocity with respect to distance:

Recalling the equation for Newton’s second law (Equation 2), and combining with Equation 13, we can now write:

And finally, rearranging Equation 14, we can write:

The derivation is complete.  On the left hand side of the equals sign, we have force multiplied by an increment of distance.  This is referred to as Work in the framework of theoretical physics.   On the right hand side, we have the kinetic energy at the end of the increment of distance minus the kinetic energy at the beginning of the distance increment.  This relationship is called the Work=Energy theorem and as we have shown, just like the Impulse=Momentum theorem, it is derived from Newton’s second law of motion.

As an aside, during the time of Newton the concept of kinetic energy was not well understood.   In fact, it wasn’t until the following century, one hundred years after Newton’s death that the relationship between work and kinetic energy was fully developed, as published in 1829 by Frenchman Gaspard-Gustave de Coriolis (Ref. 6).   

OK, SO WHAT?

Let’s take a step back and talk about why Newton’s laws of motion, and in particular Newton’s second law, the Impulse=Momentum theorem, and the Work=Energy theorem are so important.  In effect, what should we expect to get out of all this that will enlighten us about bows and arrows?  What do these expressions tell us that we didn’t already know? Well, in a nutshell, they tell us EVERYTHING about the performance of a bow and arrow and give us the means to quantify performance metrics so that comparisons can be made.   For example, we can use these equations to predict and to quantify which arrow will retain more downrange velocity, which arrow will drop more over a given range, which one will penetrate better, which bow cam or limb set is more efficient and why.   In addition to analyzing the bow and arrow directly, these equations can also guide us in both setting up proper experiments and in interpreting the resultant data correctly.   As we will see in the future articles, it will be difficult if not impossible NOT to rely on these equations when determining various performance metrics of the bow and arrow.

Finally, a key concept to understand when applying these equations is that they are a “first principles” analysis technique (Ref. 7).   That is to say, they are a fundamental analysis technique for solving problems that are idealized in such a way to allow for the analysis to be applied.    For instance, we may use the Work=Energy theorem to calculate the kinetic energy of an arrow when launched from a compound bow.   The answer will be close, but if losses due to heat, sound, friction, vibration, arrow flexing, etc. are not taken into account the answer will not match the real-world result exactly.  However, the great advantage to a first principles analysis is that it gives tremendous insight into what is physically occurring, where a more advanced analysis (such as computer simulation) may not.  A complex computer simulation may tell you that an object is behaving in a particular way, whereas a first principles analysis can tell you why an object is behaving that way.   For more details and a fuller understanding of the application of Newton’s second law, the Impulse=Momentum theorem, and Work=Energy theorem, the reader is encouraged to watch a video presentation produced by The Efficient Engineer on YouTube (Ref. 8) as well as many other fine videos on the subject. 

CONCLUSION

In this article we have looked briefly into the historical development of Newton’s laws of motion, with particular focus on the second law.   We have shown that equations expressing the Impulse=Momentum theorem and the Work=Energy theorem can be directly derived from the second law.   In future articles we will be applying these equations to examine various aspects of the bow and arrow.

References

1. https://www.gsjournal.net/Science-Journals/Communications-Mechanics%20/%20%20%20Electrodynamics/Download/4537
2. https://www.rep.routledge.com/articles/thematic/mechanics-aristotelian/v-1
3. https://www.sparknotes.com/history/european/scientificrevolution/section8/
4. https://en.wikipedia.org/wiki/Johannes_Kepler
5. https://www.physicsclassroom.com/class/1dkin/Lesson-6/Kinematic-Equations
6. https://en.wikipedia.org/wiki/Gaspard-Gustave_de_Coriolis
7. https://fs.blog/first-principles/
8. https://youtu.be/SP2hy3Uf0Ls

 

(APPENDIX)

 

UNITS CHECK: 

It is always good to do a units check on any derived mathematical formula to see if the formula contains an error.   In this section we will do sanity checks on the three formulas that we have been discussing.    Recall from Article 2 that the units for mass as measured in slugs are:

Let’s begin with by examining Newton’s second law:

Plugging units into Equation 2 gives us:

And after cancelling terms on the right hand side, we are left with:

So as expected, Newton’s second law passes a units check.

In a similar manner let’s examine Equation 4, the Impulse=Momentum theorem:

At first it may seem that the units will not be the same on both sides of the equal sign.  On the left hand side, we have units of force and time, but on the right hand side we have units of mass and velocity.  Let’s proceed and see what happens:

After cancelling terms on the right hand side we get:

So the Impulse=Momentum theorem also passes a units check.

Finally, let’s check the Work=Energy theorem, Equation 15:

On the left hand side we have units of force and distance, whereas on the right hand side we have units of mass and the velocity squared.   At first it may seem as if these units don’t agree with each other.  Let’s plug units into Equation 15 and see what happens:

After cancelling terms on the right hand side we get:

So the Work=Energy theorem also passes a units check.   We have gained confidence that our derivations are correct and we can now proceed to use the derived theorems of Impulse=Momentum and Work=Energy to gain insight into the mechanics of the bow and arrow.

 

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