variable. We can use F ¼ dP/dt to analyze variable mass systems only if we apply it
to an entire system of constant mass having parts among which there is an interchange of mass. Since Newton’s second law is valid only for constant mass systems
(Beatty 2006; Thornton 2004; Hellingman 1992), m can be taken outside the
differentiation operator by the constant factor rule in differentiation. Thus,
F ¼ m
dv
dt
¼ m a
An impulse I occurs when a force F acts over an interval of time Δt, and it is given
by
I ¼
Z
Δt
F dt
Since force is the time derivative of momentum, it follows that
I ¼ Δp ¼ m Δv
This relation between impulse and momentum is closer to Newton’s wording of
the second law. Impulse is a concept frequently used in the analysis of collisions,
impacts, and dynamic loads.
2.1.3 Third Universal Law of Motion
Newton’s original Latin text reads:
“Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum
actiones in se mutuo semper esse æquales et in partes contrarias dirigi.”
Directly translated to English, this reads:
“Law III: 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.”
(Modern translation) third law: When one body exerts a force on a second body F 12 ¼ F 21 ,
the second body simultaneously exerts a force F 21 equal in magnitude and opposite in
direction on the first body.
Newton’s explanatory comment to this law:
“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 changes the motion of the other, that body also (because of the equality of the mutual
2.1 Newton’s Universal Laws of Motion
9
to an entire system of constant mass having parts among which there is an interchange of mass. Since Newton’s second law is valid only for constant mass systems
(Beatty 2006; Thornton 2004; Hellingman 1992), m can be taken outside the
differentiation operator by the constant factor rule in differentiation. Thus,
F ¼ m
dv
dt
¼ m a
An impulse I occurs when a force F acts over an interval of time Δt, and it is given
by
I ¼
Z
Δt
F dt
Since force is the time derivative of momentum, it follows that
I ¼ Δp ¼ m Δv
This relation between impulse and momentum is closer to Newton’s wording of
the second law. Impulse is a concept frequently used in the analysis of collisions,
impacts, and dynamic loads.
2.1.3 Third Universal Law of Motion
Newton’s original Latin text reads:
“Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum
actiones in se mutuo semper esse æquales et in partes contrarias dirigi.”
Directly translated to English, this reads:
“Law III: 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.”
(Modern translation) third law: When one body exerts a force on a second body F 12 ¼ F 21 ,
the second body simultaneously exerts a force F 21 equal in magnitude and opposite in
direction on the first body.
Newton’s explanatory comment to this law:
“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 changes the motion of the other, that body also (because of the equality of the mutual
2.1 Newton’s Universal Laws of Motion
9