This document discusses Newton's second law of motion and how it relates to momentum and force. It argues that momentum, not force, is the more fundamental concept. Newton's second law, which states that force equals mass times acceleration (F=ma), can instead be expressed as the change in momentum over time. This recognizes that acceleration results from a change in velocity due to momentum transfer between objects in contact. Force is thus a result of the transfer of momentum, rather than the cause of motion or acceleration as was initially thought. Momentum is therefore the more basic concept in describing changes in motion.

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SECOND LAW
CM [011] Newton’s

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Newton’s Second Law
Acceleration motion is described by Newton’s Second Law, which says:
When a particle is acted on by an external force, it accelerates. The acceleration
is proportional to the magnitude of the applied force 𝑭𝑭 and is in the same
direction as the applied force. If the particle has a mass 𝑚𝑚 and the applied force is
represented by the vector 𝑭𝑭, then: 𝑭𝑭 = 𝑚𝑚𝒂𝒂.
State of rest Object accelerated to 𝑣𝑣Force Applied

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Constant Force and Motion
As we have said the action of changing the state of motion of an object is
tantamount to the application of force on the object. When a force is applied
horizontally to an object resting on a frictionless surface, the object moves and
attains a certain speed. Aristotle fully recognized that this force is the cause of
motion. However, he did not know about friction working on the object. Thus he
came to the observation that a constant supply of force is required to keep the
object in motion.

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Constantly Adding Force = Acceleration
In Galileo’s and Newton’s theory, if force is applied to an object constantly, there is
constant increase in momentum and thus resulted in constatnt acceleration.
Acceleration

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Investigations on Force
But this ‘force’ which is so central to Newton’s theory of motion, was not clearly explained by
Newton himself. It seems to be a kind of action causing agent which exists mysteriously in
nature which he did not explain at all. People worked hard for thousands of years trying to
find what it is: pioneer scientists wrote books on it. Notable examples are ‘Concepts of Force’
(1957) by Max Jammer (1915-2010) and ‘Forces and Fields’ (1965) by Mary Hesse (born 1924).
Hundreds of other scientists
also talk about it in thesis,
papers and textbooks. But
none can give us a
satisfactory answer to the
nature of force.

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Momentum Change
Fortunately, in Newton’s theory, there is another definition of acceleration and force.
This is the change in momentum 𝑝𝑝. Remember that acceleration is but the time
change in velocity, that is:
𝑎𝑎 =
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
If 𝐹𝐹 = 𝑚𝑚𝑚𝑚, then:
𝐹𝐹 = 𝑚𝑚𝑚𝑚 = 𝑚𝑚
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
Which ends up as:
𝐹𝐹 = 𝑚𝑚𝑚𝑚 = 𝑚𝑚
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
=
𝑑𝑑𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑚𝑚𝑚𝑚 is the momentum 𝑝𝑝 of the system. So we have:
𝐹𝐹 =
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭
=
𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪
𝒊𝒊 𝒊𝒊
𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎

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Momentum Change
Thus we have a basic nature of force as the change in momentum. This is correct
so far in view of the motion of an body by added momentum to it, equivalent to
the application of force.
Receiving momentum
and moves accordingly
Force Applied = momentum injection

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Momentum is important not only because it is the agent that choreographs
constant rectilinear motion. It is also associated with acceleration and deceleration.
When an object is at rests, the addition of momentum will spurs it on to motion; on
the other hand, when a travelling object losses momentum, it slows down and drop
to a stop.
Momentum Changes Motion

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Momentum & Acceleration
A particle at rest is one without any momentum (known previously as impetus). As
said, it is not a special state. It is only a state being without any momentum.
Particle B at rest
With velocity v = 𝑣𝑣𝐵𝐵 = 0
The state being without momentum
Palm not moving, thus no
momentum

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Mechanism of Acceleration
The particle being stricken gets the momentum from the palm and move
away with the momentum.
Particle B in motion
With velocity 𝑣𝑣 = 𝑣𝑣𝐵𝐵
Momentum is doing all
the work. The particle
now is a free one.
Palm gathering
momentum for
a push.
Particle being push thus
gaining momentum from
the palm and starts to
accelerate.
Momentum
enters body
of particle
Particle at
rest
❶ ❷ ❸

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Adding Force = Acceleration
We can also see that momentum is transferred only when objects are in contact. If we
can extend the space of contact as a large zone visible to the eye, we can see
momentum happily migrating from the pushing host to the object.
Contact zone

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Collisions – When Momenta Change Hands
The transfer of momenta may happen under many situations. But the essence is in the
duration of transfer. The momentum will be small when the transfer time in long; and
vice versa.
Particle being push thus
gaining momentum from the
palm and starts to accelerate.
The touch is soft.
In collisions, the exchange of
momenta is fast and strong’.

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Modified Second Law
So we added Part 3 of Law One into Law Two, and change force into momentum at
the same time, we have a modified version of Law Two, written as:
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑜𝑜𝑜𝑜 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡. 𝑜𝑜𝑜𝑜
𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑜𝑜𝑜𝑜 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓

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Modified Second Law
The resultant corollaries are:
i. Acceleration is the result of velocity
change:
𝑎𝑎 =
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
ii. Force is the result of momentum
change:
𝐹𝐹 = 𝑚𝑚𝑚𝑚 = 𝑚𝑚
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
=
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
iii. The direction of force is related to
the direction of the momentum transfer.
FORCE?

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Popularity of Force over Momentum
These are the main reasons why momentum is of priority in the consideration of particle
collisions. Force is a compound concept which is more convenience for handling mechanical
applications. So convenient that the origin of force is overlooked. Laymen are more familiar
with the concept of force rather than momentum and will mistake the two as the same.

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Momentum the Basic
Ingredient.
Force is the manner momentum
is transferred. So the basic
ingredient of the Second Law of
motion is also momentum.

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NEWTON'S 3RD LAW
To be continued on CM [012]:
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