Dynamics 2
MR MUBITA.D
NUCLEAR MEDICAL DOCTOR/MEDICAL
PHYSICIST
daviesmubita@gmail.com
DYNAMICS
Momentum and Newton’s laws of motion
Non-uniform motion
Linear momentum and its conservation
KINEMATICS
Kinematics is the study of motion. Movement is part of everyday experience, so it is
important to be able to analyze and predict the way in which objects move.
The behavior of moving objects is studied both graphically and through equations of
motion.
DYNAMICS
The motion of any object is governed by forces that act on the object.
This topic introduces Newton’s laws of motion, which are fundamental to understanding
the connection between forces and motion. The concept of momentum and the use of
momentum conservation to analyze interactions are also studied.
4.1 Momentum and Newton’s Laws
of Motion
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a) understand that mass is the property of a body
that resists change in motion
b) recall the relationship F = ma and solve
problems using it, appreciating
that acceleration and resultant force are always in
the same direction
c) define and use linear momentum as the
product of mass and velocity
d) define and use force as rate of change of
momentum
e) state and apply each of Newton’s laws of
motion
Newton’s Laws of Motion
Review
Background
Sir Isaac Newton (1643-1727) an English
scientist and mathematician famous for his
discovery of the law of gravity also
discovered the three laws of motion. He
published them in his book Philosophiae
Naturalis Principia Mathematica
(mathematic principles of natural
philosophy) in 1687. Today these laws are
known as Newton’s Laws of Motion and
describe the motion of all objects on the
scale we experience in our everyday lives.
“If I have ever made any valuable discoveries, it has
been owing more to patient attention, than to any
other talent.”
-Sir Isaac Newton
Newton’s Laws of Motion
1. An object in motion tends to stay
in motion and an object at rest
tends to stay at rest unless acted
upon by an unbalanced force.
2. Force equals mass times
acceleration
(F = ma).
3. For every action there is an
equal
and opposite reaction.
Newton’s First Law
An object at rest tends to stay at rest
and an object in motion tends to stay
in motion unless acted upon by an
unbalanced force.
What does this mean?
Basically, an object will “keep doing what
it was doing” unless acted on by an
unbalanced force.
If the object was sitting still, it will
remain stationary. If it was moving at a
constant velocity, it will keep moving.
It takes force to change the motion of an
object.
What is meant by unbalanced
force?
If the forces on an object are equal and opposite, they are said
to be balanced, and the object experiences no change in
motion. If they are not equal and opposite, then the forces are
unbalanced and the motion of the object changes.
Some Examples from Real Life
A soccer ball is sitting at rest. It
takes an unbalanced force of a kick
to change its motion.
Two teams are playing tug of war. They are both
exerting equal force on the rope in opposite
directions. This balanced force results in no
change of motion.
Newton’s First Law is also called
the Law of Inertia
Inertia: the tendency of an object to
resist changes in its state of motion
The First Law states that all objects
have inertia. The more mass an
object has, the more inertia it has
(and the harder it is to change its
motion).
Inertia
More Examples from Real Life
A powerful locomotive begins to pull a
long line of boxcars that were sitting at
rest. Since the boxcars are so massive,
they have a great deal of inertia and it
takes a large force to change their
motion. Once they are moving, it takes
a large force to stop them.
On your way to school, a bug
flies into your windshield. Since
the bug is so small, it has very
little inertia and exerts a very
small force on your car (so small
that you don’t even feel it).
If objects in motion tend to stay in motion,
why don’t moving objects keep moving forever?
Things don’t keep moving forever because
there’s almost always an unbalanced force
acting upon it.
A book sliding across a table slows
down and stops because of the force
of friction.
If you throw a ball upwards it will
eventually slow down and fall
because of the force of gravity.
In outer space, away from gravity and any
sources of friction, a rocket ship launched
with a certain speed and direction would
keep going in that same direction and at that
same speed forever.
Terminal Velocity
Newton’s Second Law
Force equals mass times acceleration.
F = ma
Acceleration: a measurement of how quickly an
object is changing speed.
What does F = ma mean?
Force is directly proportional to mass and acceleration.
Imagine a ball of a certain mass moving at a certain
acceleration. This ball has a certain force.
Now imagine we make the ball twice as big (double the
mass) but keep the acceleration constant. F = ma says
that this new ball has twice the force of the old ball.
Now imagine the original ball moving at twice the
original acceleration. F = ma says that the ball will
again have twice the force of the ball at the original
acceleration.
More about F = ma
If you double the mass, you double the force. If you
double the acceleration, you double the force.
What if you double the mass and the acceleration?
(2m)(2a) = 4F
Doubling the mass and the acceleration quadruples the
force.
So . . . what if you decrease the mass by half? How
much force would the object have now?
What does F = ma say?
F = ma basically means that the force of an object
comes from its mass and its acceleration.
Something very massive (high mass)
that’s changing speed very slowly (low
acceleration), like a glacier, can still
have great force.
Something very small (low mass) that’s
changing speed very quickly (high
acceleration), like a bullet, can still
have a great force. Something very
small changing speed very slowly will
have a very weak force.
Since ‘F=ma’ and ‘a = (v-u)/t’, we can have:
F = m(v-u)/t
Thus Newton’s 2nd law of motion can be
restated as:
‘the resultant force acting on a body is
proportional to the rate of change of
momentum.’
F = ∆p/∆t
For a constant mass:
F= ∆(mv)/∆t = m(∆v/∆t) = ma
NB: F is the resultant force acting on a body.
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Newton’s Third Law
For every action there is an equal and
opposite reaction.
What does this mean?
For every force acting on an object, there is an equal
force acting in the opposite direction. Right now,
gravity is pulling you down in your seat, but
Newton’s Third Law says your seat is pushing up
against you with equal force. This is why you are
not moving. There is a balanced force acting on
you– gravity pulling down, your seat pushing up.
Think about it . . .
What happens if you are standing on a
skateboard or a slippery floor and push against
a wall? You slide in the opposite direction
(away from the wall), because you pushed on
the wall but the wall pushed back on you with
equal and opposite force.
Why does it hurt so much when you stub
your toe? When your toe exerts a force on a
rock, the rock exerts an equal force back on
your toe. The harder you hit your toe against
it, the more force the rock exerts back on your
toe (and the more your toe hurts).
Action and Reaction on Different Masses
Consider you and the earth
Action: earth pulls on you
Reaction: you pull on earth
Reaction: road pushes on tire
Action: tire pushes on road
Reaction: gases push on rocket
Action: rocket pushes on gases
Review
Newton’s First Law:
Objects in motion tend to stay in motion
and objects at rest tend to stay at rest
unless acted upon by an unbalanced force.
Newton’s Second Law:
Force equals mass times acceleration
(F = ma).
Newton’s Third Law:
For every action there is an equal and
opposite reaction.
Vocabulary
Inertia:
the tendency of an object to resist changes
in its state of motion
Acceleration:
•a change in velocity
•a measurement of how quickly an object is
changing speed, direction or both
Velocity:
The rate of change of a position along
a straight line with respect to time
Force:
strength or energy
What Laws are represented?
1stlaw: Homer is large and has much
mass, therefore he has much inertia.
Friction and gravity oppose his motion.
2nd law: Homer’s mass x 9.8 m/s/s
equals his weight, which is a force.
3rd law: Homer pushes against the
ground and it pushes back.
Check yourself
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Qn. 1& 2 page 57 AS/A Level Physics
4.2 Non-uniform motion
a) describe and use the concept of weight
as the effect of a gravitational field on a
mass and recall that the weight of a body is
equal to the product of its mass and the
acceleration of free fall
b) describe qualitatively the motion of
bodies falling in a uniform gravitational field
with air resistance
Weight
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Weight of an object is the amount of the
force of gravity acting on that object.
W = mg
Terminal velocity
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This is a constant velocity that a free
falling object reaches when its weight
equals air resistance.
Examples: p. 61
 Work: p.61
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4.3 Linear momentum
and its conservation
a) state the principle of conservation of momentum
b) apply the principle of conservation of
momentum to solve simple problems, including
elastic and inelastic interactions between bodies in
both one and two dimensions (knowledge of the
concept of coefficient of restitution is not required)
c) recognize that, for a perfectly elastic collision, the
relative speed of approach is equal to the relative
speed of separation
d) understand that, while momentum of a system is
always conserved in interactions between bodies,
some change in kinetic energy may take
place
Honors Physics
Impulse and Momentum
Impulse = Momentum
Consider Newton’s 2nd Law
and the definition of
acceleration
Units of Impulse:
Ns
Units of Momentum:
Kg x m/s
Momentum is defined as “Inertia in Motion”
Impulse – Momentum Theorem
Ft  mv
IMPULSE
CHANGE IN MOMENTUM
This theorem reveals some
interesting relationships such as
the INVERSE relationship
between FORCE and TIME
mv
F
t
Impulse – Momentum Relationships
Impulse – Momentum Relationships
fT  mV
Constant
Since TIME is directly related to the
VELOCITY when the force and mass are
constant, the LONGER the cannonball is
in the barrel the greater the velocity.
Also, you could say that the force acts
over a larger displacement, thus there is
more WORK. The work done on the
cannonball turns into kinetic energy.
How about a collision?
Consider 2 objects speeding
toward each other. When they
collide......
Due to Newton’s 3rd Law the
FORCE they exert on each
other are EQUAL and
OPPOSITE.
The TIMES of impact are also
equal.
F1   F2 t1  t2
( Ft)1  ( Ft) 2
J1   J 2
Therefore, the IMPULSES of the 2
objects colliding are also
EQUAL
How about a collision?
If the Impulses are equal
then the MOMENTUMS
are also equal!
J1   J 2
p1   p2
m1v1  m2 v2
m1 (v1  vo1 )  m2 (v2  vo 2 )
m1v1  m1vo1  m2 v2  m2 vo 2
p
before
  p after
m1vo1  m2 vo 2  m1v1  m2 v2
Momentum is conserved!
The Law of Conservation of Momentum: “In the
absence of an external force (gravity, friction),
the total momentum before the collision is
equal to the total momentum after the
collision.”
po (truck )  mvo  (500)(5)  2500kg * m / s
po ( car )  (400)(2)  800kg * m / s
po (total )  3300kg * m / s
ptruck  500 * 3  1500kg * m / s
pcar  400 * 4.5  1800kg * m / s
ptotal  3300kg * m / s
Types of Collisions
A situation where the objects DO NOT
STICK is one type of collision
Notice that in EACH case, you have TWO objects BEFORE and AFTER the
collision.
A “no stick” type collision
Spbefore
=
Spafter
m1vo1  m2 vo 2  m1v1  m2v2
(1000)(20)  0  (1000)(v1 )  (3000)(10)
 10000
v1 
-10 m/s
 1000v1
Types of Collisions
Another type of collision is one where the
objects “STICK” together. Notice you
have TWO objects before the collision
and ONE object after the collision.
A “stick” type of collision
Spbefore
=
Spafter
m1vo1  m2vo 2  mT vT
(1000)(20)  0  (4000)vT
 4000vT
20000
vT 
5 m/s
The “explosion” type
This type is often referred to as
“backwards inelastic”. Notice you have
ONE object ( we treat this as a
SYSTEM) before the explosion and
TWO objects after the explosion.
Backwards Inelastic - Explosions
Suppose we have a 4-kg rifle
loaded with a 0.010 kg bullet.
When the rifle is fired the
bullet exits the barrel with a
velocity of 300 m/s. How fast
does the gun RECOIL
backwards?
Spbefore
mT vT
Spafter
=
 m1v1  m2 v2
(4.010)(0)  (0.010)(300)  (4)(v2 )
0
 3  4v2
v2

-0.75 m/s
Collision Summary
Sometimes objects stick together or blow apart.
In this case, momentum is ALWAYS conserved.
p
before
  p after
m1v01  m2 v02  m1v1  m2 v2
When 2 objects collide and DON’T stick
m1v01  m2 v02  mtotal vtotal
When 2 objects collide and stick together
mtotal vo (total )  m1v1  m2 v2
When 1 object breaks into 2 objects
Elastic Collision = Kinetic Energy is Conserved
Inelastic Collision = Kinetic Energy is NOT Conserved
Elastic Collision
KE car ( Before)  1 mv 2  0.5(1000)(20) 2  200,000 J
2
KEtruck ( After)  0.5(3000)(10) 2  150,000 J
KE car ( After)  0.5(1000)(10) 2  50,000 J
Since KINETIC ENERGY is conserved during the collision we call this an
ELASTIC COLLISION.
Inelastic Collision
KEcar ( Before)  1 mv 2  0.5(1000)(20) 2  200,000 J
2
KEtruck / car ( After)  0.5(4000)(5) 2  50,000 J
Since KINETIC ENERGY was NOT conserved during the collision we call this an
INELASTIC COLLISION.
Example
How many objects do I have before the collision?
2
How many objects do I have after the collision?
1
Granny (m=80 kg) whizzes
around the rink with a velocity
of 6 m/s. She suddenly collides
with Ambrose (m=40 kg) who
is at rest directly in her path.
Rather than knock him over,
she picks him up and continues
in motion without "braking."
Determine the velocity of
Granny and Ambrose.

pb   pa
m1vo1  m2 vo 2  mT vT
(80)(6)  (40)(0)  120vT
vT  4 m/s
Collisions in 2 Dimensions
The figure to the left shows a
collision between two pucks
on an air hockey table. Puck A
has a mass of 0.025-kg and is
vA
moving along the x-axis with a
vAsinq
velocity of +5.5 m/s. It makes
a collision with puck B, which
has a mass of 0.050-kg and is
vAcosq
initially at rest. The collision is
NOT head on. After the
collision, the two pucks fly
vBcosq
vBsinq apart with angles shown in
vB
the drawing. Calculate the
speeds of the pucks after the
collision.
Collisions in 2 dimensions
p
ox
  px
m AvoxA  mB voxB  m AvxA  mB vxB
(0.025)(5.5)  0  (.025)(v A cos 65)  (.050)(vB cos 37)
vA
vAsinq
vAcosq
vBcosq
vB
0.1375  0.0106vA  0.040vB
p
oy
  py
0  mAv yA  mB v yB
vBsinq
0  (0.025)(v A sin 65)  (0.050)(vB sin 37)
0.0300vB  0.0227v A
vB  0.757v A
Collisions in 2 dimensions
0.1375  0.0106vA  0.040vB
vB  0.757v A
0.1375  0.0106v A  (0.050)(0.757v A )
0.1375  0.0106v A  0.03785v A
0.1375  0.04845v A
v A  2.84m / s
vB  0.757(2.84)  2.15m / s
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Check yourself:
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Examination style questions
p.67
TE
HE
ND