Chapter Four Notes:
Newton’s Second Law of Motion – Linear Motion
Newton's Second Law
Newton's first law of motion predicts the
behavior of objects for which all existing forces
are balanced. The first law - sometimes referred
to as the law of inertia - states that if the forces
acting upon an object are balanced, then the
acceleration of that object will be 0 m/s/s.
Objects at equilibrium (the condition in which all
forces balance) will not accelerate. According to
Newton, an object will only accelerate if there is a
net or unbalanced force acting upon it. The
presence of an unbalanced force will accelerate
an object - changing either its speed, its
direction, or both its speed and direction.
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Newton's second law of motion pertains to the behavior of objects for
which all existing forces are not balanced. The second law states that
the acceleration of an object is dependent upon two variables - the
net force acting upon the object and the mass of the object. The
acceleration of an object depends directly upon the net force acting
upon the object, and inversely upon the mass of the object. As the
force acting upon an object is increased, the acceleration of the object
is increased. As the mass of an object is increased, the acceleration of
the object is decreased.
Newton's second law of motion can be formally stated as follows:
The acceleration of an object as produced by a net force is directly
proportional to the magnitude of the net force, in the same direction as
the net force, and inversely proportional to the mass of the object.
This verbal statement can be expressed in equation form as follows:
a = Fnet / m
The above equation is often rearranged to a more familiar form as
shown below. The net force is equated to the product of the mass
times the acceleration.
Fnet = m * a
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In this entire discussion, the emphasis has been
on the net force. The acceleration is directly
proportional to the net force; the net force equals
mass times acceleration; the acceleration in the
same direction as the net force; an acceleration is
produced by a net force. The NET FORCE. It is
important to remember this distinction.
Do not use the value of merely "any 'ole force" in
the above equation. It is the net force which is
related to acceleration. As discussed in another
lesson, the net force is the vector sum of all the
forces. If all the individual forces acting upon an
object are known, then the net force can be
determined.
NET FORCE 
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Consistent with the equation on previous
slide, a unit of force is equal to a unit of mass
times a unit of acceleration. By substituting
standard metric units for force, mass, and
acceleration into the above equation, the
following unit equivalency can be written.
The definition of the standard metric unit of
force is stated by the above equation. One
Newton is defined as the amount of force
required to give a 1-kg mass an acceleration
of 1 m/s/s (1ms-2).

The Fnet = m • a equation is often used in
algebraic problem-solving. The table below
can be filled by substituting into the equation
and solving for the unknown quantity. Try it
yourself and then click the mouse to view the
answers.
Net Force
(N)
Mass
(kg)
Acceleration
(m/s/s)
1
10
2
5
2
20
2
10
3
20
4
5
4
10
2
5
5
10
1
10
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The numerical information in the table above demonstrates some
important qualitative relationships between force, mass, and
acceleration. Comparing the values in rows 1 and 2, it can be seen that a
doubling of the net force results in a doubling of the acceleration (if
mass is held constant). Similarly, comparing the values in rows 2 and 4
demonstrates that a halving of the net force results in a halving of the
acceleration (if mass is held constant). Acceleration is directly
proportional to net force.
Furthermore, the qualitative relationship between mass and acceleration
can be seen by a comparison of the numerical values in the above table.
Observe from rows 2 and 3 that a doubling of the mass results in a
halving of the acceleration (if force is held constant). And similarly, rows
4 and 5 show that a halving of the mass results in a doubling of the
acceleration (if force is held constant). Acceleration is inversely
proportional to mass.
The analysis of the table data illustrates that an equation such as Fnet =
m*a can be a guide to thinking about how a variation in one quantity
might effect another quantity. Whatever alteration is made of the net
force, the same change will occur with the acceleration. Double, triple or
quadruple the net force, and the acceleration will do the same. On the
other hand, whatever alteration is made of the mass, the opposite or
inverse change will occur with the acceleration. Double, triple or
quadruple the mass, and the acceleration will be one-half, one-third or
one-fourth its original value.
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In conclusion, Newton's second law provides the
explanation for the behavior of objects upon
which the forces do not balance. The law states
that unbalanced forces cause objects to
accelerate with an acceleration which is directly
proportional to the net force and inversely
proportional to the mass.
 Acceleration:
An often confused quantity, acceleration has a
meaning much different than the meaning
associated with it by sports announcers and
other individuals. The definition of acceleration
is:
Acceleration is a vector quantity which is defined
as the rate at which an object changes its
velocity. An object is accelerating if it is changing
its velocity (Either it’s speed and/or direction).
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An object is moving if its position relative to
a fixed point is changing.
Speed is how fast an object is moving. You
can calculate the speed of an object by
dividing the distance covered by time.
 Speed = distance/time
 SI Units: meters per second (m/s)

Instantaneous Speed
◦ The speed of an object at any given instant!
◦ You can tell the speed of a car at any instant by
looking at the speedometer.
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Average Speed
◦ Total distance covered divided by the total time:
 Average speed = total distance /time interval
◦ Ea: if we traveled 240 kilometers in 4 hours
◦ Average speed = 240 km/4 h = 60 km/h
◦ Simple variation of this equation gives:
◦ Total distance traveled = average speed X travel time
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Speed is a description of how fast an object
moves; velocity is how fast and in what
direction it moves.
Constant Velocity:
◦ Both speed and direction remain constant!

Changing Velocity:
◦ Either the speed or the direction (or both) change.
Then you have changing velocity!
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You can calculate the acceleration of an object
by dividing the change in its velocity by time.
 Acceleration = Δ velocity / time interval
 (Δ stands for “change in”)
Acceleration can be either positive (increasing
speed) or negative (decreasing speed). We
often
call
the
negative
acceleration,
deceleration.
Acceleration also applies to a change in
direction!
If we change speed, direction or
both, we change velocity and we accelerate!
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An object moving under the influence of the
gravitational force only is said to be in free
fall.
The elapsed time
is the time that has
elapsed, or passed, since the beginning of
any motion, in this case free fall.
The acceleration due to gravity
on the
surface of earth is given by the following
value in this text:
 g = 10 m/s2

Therefore, in free fall an object increases in
speed by an additional 10 meters per second.
Free Fall Speeds of Objects
Elapsed Time
(seconds)
Instantaneous Speed
(meters/second)
0
0
1
10
2
20
3
30
4
40
5
50
t
10t
V = gt
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Rising Objects:
◦ An object thrown straight up, will slow down the
same way a free fall object speeds up, 10 m/s each
second. If thrown up with a speed of 30 m/s, will
have the following speeds at each second,
assuming up is positive and down is negative.
Time (sec)
Speed (m/s) Time (sec)
Speed (m/s)
3
0
3
0
2
10
4
-10
1
20
5
-20
0
30
6
-30
7
-40
8
-50
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For a freely falling object, for each second of
free fall, an object falls a greater distance
than it did in the previous second. You need
to calculate this distance by taking the
average speed over the interval and
multiplying it by 1 second.
During the first second: Initial speed = 0 m/s,
final speed = 10 m/s, so average speed is
(final speed + initial speed)/2.
[(10 m/s + 0 m/s) /2]/1sec = 5 meters
during 1st sec.
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During 2nd second: Initial speed = 10 m/s, final
speed = 20 m/s. Therefore:
[(20 m/s + 10 m/s)/2]/1 sec = 15 meters.
During 3rd second: Initial speed = 20 m/s, final
speed = 30 m/s. Therefore:
[(30 m/s + 20 m/s)/2]/1 sec = 25 meters.
 NOW:
the total distance since the beginning becomes:
(1 sec):
0+ 5 =
5 meters
(2 sec):
5 + 15 = 20 meters
(3 sec): 20 + 25 = 45 meters
(4 sec): 45 + 35 = 80 meters
(5 sec): 80 + 45 = 125 meters
During tth sec: Total Distance = ½gt2 meters.
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Speed-Versus-Time
◦ On a speed-versus-time graph, the slope represents speed
per time, or acceleration.
◦
◦
◦
◦
◦
◦
◦
Speed
(m/s)
Slope =
rise/run =
(acceleration)
Time (s)
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Distance-Versus-Time
◦ On a distance-versus-time graph, the slope
represents distance per time, or instantaneous
speed.
◦ Distance
◦
(m)
◦
◦
◦
◦
◦
slope =
rise/run =
speedinst
0
1
2
3
Time (s)
4
5
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Air resistance noticeably slows the motion of
things with large surface areas like falling
feathers or pieces of paper. But air resistance
less noticeably affects the motion of more
compact objects like stones and baseballs.
A classic demonstration of this is called “the
feather and guinea demonstration. It involves
a tube with both items in it. When the air is
removed, both items fall at the same speed.
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Some of the confusion that occurs in
analyzing the motion of falling objects comes
about from mixing up “how fast” with “how
far.”
When we wish to specify how fast, the
equation is:
v = gt
When we wish to specify how far, the
equation is:
d = (1/2)gt2
Acceleration is the rate at which velocity itself
changes. [the rate of a rate]