Chapter 2: Motion, Forces,
& Newton’s Laws
Brief Overview of the Course
“Point” Particles & Large Masses
• Translational Motion = Straight line motion.
– Chapters 2,3,4,6,7
• Rotational Motion = Moving (rotating) in a circle.
– Chapters 5,8
• Oscillations = Moving (vibrating) back & forth on same path.
– Chapter 11
Continuous Media
• Waves, Sound
– Chapters 11,12
• Fluids = Liquids & Gases
– Chapter 10
THE COURSE
THEME IS
NEWTON’S LAWS
OF MOTION!!
Conservation Laws
• Energy, Momentum, Angular Momentum
• Newton’s Laws expressed in other forms!
Chapter 2 Topics
• Reference Frames & Displacement
• Average Velocity
• Instantaneous Velocity
• Acceleration
• Motion at Constant Acceleration
• Solving Problems
• Freely Falling Objects
Terminology
• Mechanics = Study of objects in motion.
– 2 parts to mechanics.
• Kinematics =
Description of HOW objects move.
– Chapters 2 & 3
• Dynamics = WHY objects move.
– Introduction of the concept of FORCE.
– Causes of motion, Newton’s Laws
– Most of the course from Chapter 4 & beyond.
• For a while, Assume Ideal Point Masses (no
physical size). Later, extended objects with size.
Terminology
• Translational Motion =
Motion with no rotation.
• Rectilinear Motion =
Motion in a straight line path.
Reference Frames & Displacement
• Every measurement must be made with respect to a
reference frame. Usually, the speed is relative to the Earth.
• For example, if you are sitting on a train & someone walks down
the aisle, the person’s speed with respect to the train is a few km/hr,
at most. The person’s speed with respect to the ground is much higher.
• Specifically, if a person walks towards the front of a train at
5 km/h (with respect to the train floor) & the train is moving
80 km/h with respect to the ground. The person’s speed,
relative to the ground is 85 km/h.
• When specifying speed, always specify
the frame of reference unless its
obvious (“with respect to the Earth”).
• Distances are also measured in a
reference frame.
• When specifying speed or distance, we
also need to specify DIRECTION.
Coordinate Axes
• Usually, we define a reference frame using
a standard coordinate axes. (But the choice
of reference frame is arbitrary & up to us, as
we’ll see later!)
• 2 Dimensions (x,y)
- ,+
+,+
• Note, if it is convenient,
we could reverse + & - !
+,-,A standard set of xy
(Cartesian or rectangular)
coordinate axes
Coordinate Axes
• 3 Dimensions (x,y,z)
First Octant
• Define direction using these.
Displacement & Distance
Distance traveled by an object
 Displacement of the object!
Here,
Distance = 100 m.
Displacement
= 40 m East.
• Displacement  Δx  Change in position
of an object. Δx is a vector (magnitude & direction).
• Distance is a scalar (magnitude).
Displacement
t1

t2

 times
x1 = 10 m, x2 = 30 m
Displacement  ∆x = x2 - x1 = 20 m
• ∆  Greek letter “delta” meaning “change in”
• The arrow represents the displacement (meters).
x1 = 30 m, x2 = 10 m
Displacement  ∆x = x2 - x1 = - 20 m
• Displacement is a VECTOR
Vectors and Scalars
• Many quantities in physics, like displacement,
have a magnitude and a direction. Such
quantities are called VECTORS.
– Other quantities which are vectors: velocity,
acceleration, force, momentum, ...
• Many quantities in physics, like distance,
have a magnitude only. Such quantities are
called SCALARS.
– Other quantities which are scalars: speed,
temperature, mass, volume, ...
• The Text uses BOLD letters to denote
vectors.
• I usually denote vectors with arrows
over the symbol.
• In one dimension, we can drop the arrow
and remember that a + sign means the
vector points to right & a minus sign means
the vector points to left.
Average Velocity
Average Speed  (Distance traveled)/(Time taken)
A Scalar
A Vector
Average Velocity  (Displacement)/(Time taken)
• Velocity: Both magnitude & direction describing
how fast an object is moving. It is a VECTOR.
• Speed: Magnitude only describing how fast
an object is moving. It is a SCALAR.
Units of both are distance/time = m/s
Average Velocity & Average Speed
• Consider the displacement from before. Suppose
that the person does the whole trip in 70 s.
Average Speed = (100 m)/(70 s) = 1.4 m/s
Average Velocity = (40 m)/(70 s) = 0.57 m/s
General Case
t1

t2  times

Bar denotes
average
∆x = x2 - x1 = displacement
∆t = t2 - t1 = elapsed time
Average Velocity
(x2 - x1)/(t2 - t1)
Velocity and Position
• Consider the case where
the position vs. time curve
is as shown in the figure.
•In general,
The Average Velocity is
the slope of the line
segment that connects
the positions at the
beginning & end of the
time interval.
Example 2.1: Velocity of a Bicycle
• Calculate the average velocity from t = 2.0 s to 3.0 s.
• The displacement is Δx = 12 m – 5 m = 7 m. So
vave = (Δx)/(Δt) = (7 m)/(1 s) = 7 m/s
Instantaneous Velocity
•Average velocity doesn’t tell
us anything about details
during the time interval.
• To look at some of the details,
smaller time intervals are needed
The slope of the curve at the
time of interest will give the
instantaneous velocity at that
time.
Instantaneous Velocity
• Instantaneous Velocity  The velocity at any
instant of time  The Average Velocity over an
infinitesimally short time.
• Mathematically, the Instantaneous Velocity is
formally defined as:
 the ratio considered as a whole for
infinitesimally small ∆t.
• Mathematicians call this a derivative.
• Do not set ∆t = 0 because ∆x = 0 then & 0/0 is undefined!
 Instantaneous velocity v
Instantaneous Velocity  Velocity
at any instant of time.
• Mathematically, instantaneous velocity is:
• Mathematicians call this a derivative.
 Instantaneous Velocity
v ≡ Time Derivative of Displacement x
These graphs show
(a) Constant Velocity 
Instantaneous Velocity
= Average Velocity
and
(b) Varying Velocity 
Instantaneous Velocity
 Average Velocity
The instantaneous velocity is the average
velocity in the limit as the time interval
becomes infinitesimally short.
Ideally, a speedometer would
measure instantaneous
velocity; in fact, it measures
average velocity, but over a
very short time interval.
Acceleration
• Velocity can change with time. An object with
velocity that is changing with time is said to be
accelerating.
• Definition: Average acceleration = ratio of
change in velocity to elapsed time.
a
= (v2 - v1)/(t2 - t1)
Acceleration is a vector.
• Instantaneous acceleration
–
a
–
• Units: velocity/time = distance/(time)2 = m/s2
Graphical Analysis of Velocity
(Example 2.3)
• To find the velocity
graphically:
– Find the slope of the line
tangent to the x-t graph
at the appropriate times
– For the average velocity
for a time interval, find the
slope of the line
connecting the two times
Example: Average Acceleration
A
A car accelerates along a straight road
from rest to 90 km/h in 5.0 s. Find the
magnitude of its average acceleration.
Note: 90 km/h = 25 m/s
a=
Example: Average Acceleration
A car accelerates along a straight road
from rest to 90 km/h in 5.0 s. Find the
magnitude of its average acceleration.
Note: 90 km/h = 25 m/s
a=
= (25 m/s – 0 m/s)/5 s = 5 m/s2
Conceptual Question
Velocity & Acceleration are both vectors.
Are the velocity and the acceleration
always in the same direction?
Conceptual Question
Velocity & Acceleration are both vectors.
Are the velocity and the acceleration
always in the same direction?
NO!!
If the object is slowing down, the acceleration
vector is in the opposite direction of the
velocity vector!
Example: Car Slowing Down
A car moves
a = to= the
(v2 –right
v1)/(t2on
– t1a) =
straight
(5 m/s –highway
15 m/s)/(5s(positive
– 0s)
x-axis). The driver puts on the brakes.
The initial velocity
2
a = - 2.0 m/s
(when the driver hits the brakes) is v1 = 15.0 m/s. It takes
t = 5.0 s to slow down to v2 = 5.0 m/s. Calculate the car’s
average acceleration.
The same car is moving
to the left instead of to
the right. Still assume
positive x is to the right.
The car is decelerating
& the initial & final
velocities are the same
as before. Calculate the
average acceleration
now.
Deceleration
• “Deceleration”: A word which means “slowing down”.
We try to avoid using it in physics. Instead (in one
dimension), we talk about positive & negative acceleration.
• This is because (for one dimensional motion) deceleration does
not necessarily mean the acceleration is negative!
Conceptual Question
Velocity & Acceleration are both vectors.
Is it possible for an object to have a zero
acceleration and a non-zero velocity?
Conceptual Question
Velocity & Acceleration are both vectors.
Is it possible for an object to have a zero
acceleration and a non-zero velocity?
YES!!
If the object is moving at a constant velocity,
the acceleration vector is zero!
Conceptual Question
Velocity & acceleration are both vectors.
Is it possible for an object to have a zero
velocity and a non-zero acceleration?
Conceptual Question
Velocity & acceleration are both vectors.
Is it possible for an object to have a zero
velocity and a non-zero acceleration?
YES!!
If the object is instantaneously at rest (v = 0)
but is either on the verge of starting to
move or is turning around & changing
direction, the velocity is zero, but the
acceleration is not!
One-Dimensional Kinematics Examples
As already noted, the instantaneous acceleration
is the average acceleration in the limit as the time
interval becomes infinitesimally short.
The instantaneous
slope of the velocity
versus time curve is
the instantaneous
acceleration.
Example: Analyzing with graphs: The figure shows the
velocity v(t) as a function of time for 2 cars, both accelerating
from 0 to 100 km/h in a time 10.0 s. Compare (a) the average
acceleration; (b) the instantaneous acceleration; & (c) the total
distance traveled for the 2 cars.
Solution: (a) Ave. acceleration:
a = Both have the same ∆v & the
same ∆t so a is the same for both.
(b) Instantaneous acceleration: a =
slope of tangent to v vs t curve. For
about the first 4 s, curve A is steeper
than curve B, so car A has greater a
than car B for times t = 0 to t = 4 s.
Curve B is steeper than curve A, so
car B has greater a than car A for
times t greater than about t = 4 s.
(c) Total distance traveled: Except for t = 0 & t = 10 s, car A is moving
faster than car B. So, car A will travel farther than car B in the same time.
Example (for you to work!) : Calculating Average Velocity & Speed
Problem: Use the figure & table to find the displacement & the
average velocity of the car between positions (A) & (F).
Example (for you to work!) : Graphical Relations between x, v, & a
Problem: The position of an object moving along the x axis varies
with time as in the figure. Graph the velocity versus time and
acceleration versus time curves for the object.