Chapter 2
Motion Along a Straight Line
© 2016 Pearson Education Inc.
What is the motion of an Object?
A change of an object’s position with time.
What are the basic types of motion?
1.
2.
3.
4.
Linear Motion
Circular Motion
Projectile Motion
Rotational Motion
2
3
Learning Goals for Chapter 2
Looking forward at …
• how the ideas of displacement and average velocity help us
describe straight-line motion.
• the meaning of instantaneous velocity; the difference
between velocity and speed.
• how to use average acceleration and instantaneous
acceleration to describe changes in velocity.
• how to solve problems in which an object is falling freely
under the influence of gravity alone.
• how to analyze straight-line motion when the acceleration is
not constant.
When analyzing motion , we consider measurements with respect to inertial
reference frame, which non accelerating reference frame. Example
ground/earth.
© 2016 Pearson Education Inc.
Motion along a straight line
(Kinematics in I-D)
• Kinematics : the branch of mechanics that
deals with pure motion, without reference to
the masses or forces involved in it.
• The motion of an object is described by its
position, displacement, velocity, and
acceleration.
• In one dimension, these quantities are
represented by x, vx , and ax.
5
2.1 & 2.2 Displacement, Time, and Velocity
Position
• Use the symbol x to denote position vector
0
x
X axis
– Note: we leave the vector arrows off for 1d
vectors
– All position vectors are measured relative to the
origin of the coordinate system, which can be
chosen arbitrarily
– Vector x can be positive or negative (its magnitude is
again always positive, though)
6
Position vector is a function of time
Notation for the time-dependent position vector: x(t)
Notation: Vector x at some specific time t1: x(t1)=x1
X(t)
X(t1)
t
t1
7
Linear Displacement
• Displacement is a vector measure of the interval between two
locations measured along the shortest path connecting them.
∆𝑥 = 𝑥2 − 𝑥1 = 𝑥 𝑡2 − 𝑥(𝑡1 )
• Displacement is a vector, just like position; can be negative
• Displacement is independent of choice of origin of coordinate
system (unlike position)
• Displacement for going from point b to point a is exactly the negative
of going from point a to point b:
𝑥𝑏𝑎 = (𝑥𝑏 − 𝑥𝑎 ) = −(𝑥𝑎 −𝑥𝑏 ) = − 𝑥𝑎𝑏
8
Distance
• Measure of the interval between two locations
measured along the actual path connecting them.
• Distance is a scalar (has only magnitude),
displacement is a vector (has both magnitude and
direction)
9
QuickCheck 2.1
An ant zig-zags back and forth on a picnic table as shown.
The ant’s distance traveled and displacement are
A.
B.
C.
D.
E.
50 cm and 50 cm.
30 cm and 50 cm.
50 cm and 30 cm.
50 cm and –50 cm.
50 cm and –30 cm.
10
QuickCheck 2.1
An ant zig-zags back and forth on a picnic table as shown.
The ant’s distance traveled and displacement are
A.
B.
C.
D.
E.
50 cm and 50 cm.
30 cm and 50 cm.
50 cm and 30 cm.
50 cm and –50 cm.
50 cm and –30 cm.
11
Linear Velocity
• Average linear velocity
– Displacement (∆𝑟) 𝑜𝑣𝑒𝑟 given time interval (∆𝑡) during which the
displacement occurs.
𝑣av =
𝑟2 − 𝑟1
𝑡2 −𝑡1
=
∆𝑟
∆𝑡
SI units: m/s
Remember you are working with vectors so you CANNOT just take
the difference of position to calculate the average velocity if the
position vectors are in different directions.
– This is the average velocity in x direction.
𝑣𝑎𝑣,𝑥 =
𝑥2 −𝑥1
𝑡2 −𝑡1
=
∆𝑥
∆𝑡
12
Instantaneous linear velocity
The instantaneous velocity is the object’s velocity at a single instant of
time t.
𝑣 𝑡 =
∆𝑟
lim
Δ𝑡→0 ∆𝑡
=
𝑑𝑟
𝑑𝑡
Obtained in the limit that the time interval for the averaging procedure
approaches 0
How do you calculate the instantaneous velocity at particular time (at t3) from
position vs time graph? (Hint: Find the slope of a point on a curve)
Draw a straight line that touches a curve at single point ( in this example t3 )
and it does not cross through it. ( we call it tangent). Calculate the slope of
that line. That will be the instantaneous velocity at time t3
13
Linear Speed (v)
• How fast an object is going
• rate of change of distance with respect to
time.
• Velocity is a vector, speed a scalar!
𝑣=
𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑒𝑑
𝑔𝑖𝑣𝑒𝑛 𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
l
v
t
SI units: m/s
14
2.3 Average and Instantaneous Acceleration
Average Linear Acceleration
• Average acceleration
– Change in velocity over a given time interval
𝑎av =
𝑣2 − 𝑣1
𝑡2 −𝑡1
𝑎𝑎𝑣−𝑥 =
=
𝑣2𝑥 −𝑣1𝑥
𝑡2 −𝑡1
∆𝑣
∆𝑡
=
SI units: m/s2
∆𝑣𝑥
∆𝑡
15
Instantaneous Linear Acceleration
Obtained in the limit that the time interval for the averaging
procedure approaches 0
𝑎 𝑡 =
∆𝑣
lim
Δ𝑡→0 ∆𝑡
=
𝑑𝑣
𝑑𝑡
16
MOTION DIAGRAMS
• Motion diagrams are a pictorial description of an
object in motion. They show an object's position
and velocity at the start, end, and several spots in
the middle, along with acceleration (if any).
Visualizing Motion ( For simplicity we use particle
model) use position of object for same amount of time
elapses between each image and the next.
17
Making
a Motion
Diagram DIAGRAM
MAKING
A MOTION
• Consider a movie of a
moving object.
• A movie camera takes
photographs at a fixed rate
(i.e., 30 photographs every
second).
• Each separate photo is
called a frame.
• The car is in a different
position in each frame.
• Shown are four frames in a
filmstrip.
18
Making a Motion Diagram
MAKING A MOTION DIAGRAM
• Cut individual frames of the film strip apart.
• Stack them on top of each other.
• This composite photo shows an object’s position at
several equally spaced instants of time.
• This is called a motion diagram.
Reproduce the motion diagram of the above description using particle
model
19
Examples
of Motion
EXAMPLES
OF Diagrams
MOTION DIAGRAMS
• An object that has a single position in a motion diagram is at
rest.
• Example: A stationary ball on the ground.
• An object with images that are equally spaced is moving
with constant speed.
• Example: A skateboarder rolling down the sidewalk.
20
Examples
of Motion
EXAMPLES
OF Diagrams
MOTION DIAGRAMS
• An object with images that have increasing distance
between them is speeding up.
• Example: A sprinter starting the 100 meter dash.
• An object with images that have decreasing distance
between them is slowing down.
• Example: A car stopping for a red light.
21
Uniform Motion
UNIFORM LINEAR MOTION
• If you drive your car at a
perfectly steady 60 mph,
this means you change your
position by 60 miles for every
time interval of 1 hour.
• Uniform motion is when equal
displacements occur during any
successive
equal-time intervals.
Riding steadily over level ground is a good
example of uniform motion.
UNIFORM LINEAR MOTION
m Motion
• An object’s motion is
uniform if and only if its
position-versus-time graph
is a straight line.
The position in x direction as a function of
time graph
Motion diagrams
• Here is a motion diagram of the particle in the
previous x-t graph.
A graph of velocity in x direction as
a function of time
Motion diagrams
• Here is the motion diagram for the particle in
the previous vx-t graph.
A vx-t graph
Motion diagrams
• Here is the motion diagram for the particle in
the previous vx-t graph.
2.4 Motion with Constant Acceleration
30
An Equation for the position x as a function of time when
the acceleration in the x direction is a constant
31
The relationship among position, velocity and acceleration
for a constant acceleration
32
2.6 Freely falling bodies
http://www.grc.nasa.gov/WWW/K-12/airplane/ffall.html
The remarkable observation that all free falling objects fall with the same acceleration was first proposed by Galileo Galilei nearly 400 years ago. Galileo conducted
experiments using a ball on an inclined plane to determine the relationship between the time and distance traveled. He found that the distance depended on the
square of the time and that the velocity increased as the ball moved down the incline. The relationship was the same regardless of the mass of the ball used in the
experiment. The experiment was successful because he was using a ball for the falling object and the friction between the ball and the plane was much smaller than
the gravitational force. He also used a very shallow incline, so the velocity was small and the drag on the ball was very small compared to the gravitational force.33
Freely falling bodies
• Free fall is the motion of
an object under the
influence of only gravity.
• In the figure, a strobe light
flashes with equal time
intervals between flashes.
• The velocity change is the
same in each time interval,
so the acceleration is
constant.
© 2016 Pearson Education Inc.
A freely falling coin
• If there is no air resistance, the downward
acceleration of any freely falling object is
g = 9.80 m/s2 = 32 ft/s2.
Calculus Reminder (from Math Primer)
d n
x  nx n 1
dx
d
sin ax   a cos ax 
dx
• Polynomials:
• Trig functions:
• Exponential, log:
• Product rule:
• Chain rule:
d ax
e  aeax
dx
d
1
ln ax  
dx
x
d
 df (x) 
 dg(x) 
 f (x)g(x)  
 g(x)  f (x) 

dx
dx
dx
dy dy du
y (u ( x)) 

dx du dx
36
2.6 Velocity and position by integration
• The acceleration of a car is not always constant.
We can divide time interval between times
𝑡1 𝑎𝑛𝑑 𝑡2 into many smaller subintervals calling a
typical one , ∆𝑡
∆𝑣𝑥 = 𝑎𝑎𝑣−𝑥 ∆𝑡 , ∆𝑣𝑥 is the area of the shaded
strip with height 𝑎𝑎𝑣−𝑥 and width ∆𝑡
Review 2.6 Velocity and position by integration
( Page 53-54)
The total x velocity change is represented graphically by the total
area under the 𝑎𝑥 𝑣𝑠 𝑡 curve between vertical lines between
𝑡1 𝑎𝑛𝑑 𝑡2 .
In the limit that all ∆𝑡 ′ 𝑠 become very small and they become
very large in numbers, the value 𝑎𝑎𝑣−𝑥 for the interval from any
time 𝑡 to 𝑡 + ∆𝑡 approaches the instantaneous acceleration 𝑎𝑥 at
time 𝑡.
∆𝑣𝑥 (𝑡)
𝑑𝑣𝑥
𝑎𝑥 𝑡 = lim ∆𝑡 → 0
=
∆𝑡
𝑑𝑡
The motion may be integrated over many small time
intervals to give
and
Time limit is from 0 𝑡𝑜 𝑡𝑖𝑚𝑒 𝑡
𝑤ℎ𝑒𝑛 𝑡 = 0 𝑠, 𝑡ℎ𝑒 𝑣𝑥 𝑡 = 0 = 𝑣𝑜𝑥 𝑎𝑛𝑑 𝑥 𝑡 = 0 = 𝑥𝑜
𝑤ℎ𝑒𝑛 𝑡 = 𝑡 𝑠, 𝑡ℎ𝑒 𝑣𝑥 𝑡 = 𝑡 = 𝑣𝑥 𝑎𝑛𝑑 𝑥 𝑡 = 𝑡 = 𝑥
38
(𝑡
𝑥 = 𝑥𝑜 +
𝑣𝑜𝑥 + 𝑎𝑥 𝑡) 𝑑𝑡
𝑡
𝑥 = 𝑥𝑜 +
0
𝑡
𝑣𝑜𝑥 𝑑𝑡 +
𝑜
𝑎𝑥 𝑡 𝑑𝑡
𝑜
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑠 𝑤ℎ𝑒𝑛 𝑡𝑖𝑚𝑒 𝑡 = 0 𝑣𝑜𝑥 𝑎𝑛𝑑 𝑎𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑥 = 𝑥𝑜 + 𝑣𝑜𝑥 𝑡 + 𝑎𝑥 𝑡 2
39
Please go over examples 2.1 and 2.6 in your
textbook and try to answer multiple choice
questions listed in next few slides.
40
Example: Velocity (1)
• Between 0 and 10 s, the position vector of a car on the road is given by
x(t)  17.2 m  (10.1 m)(t / s)+(1.1 m)(t /s)2
Question: What is its velocity vector at any given time t?
Answer: Take derivative
dx
v(t) 
dt
d

17.2 m  (10.1 m)(t / s)+(1.1 m)(t/s)2
dt
d
d
d

17.2 m 
(10.1 m)(t / s) 
(1.1 m)(t/s)2
dt
dt
dt
 0  10.1 m/s+2  (1.1 m)(t / s 2 )








 10.1 m/s+(2.2 m/s)(t / s)
41
Example: Velocity (2)
x(t)  17.2 m  (10.1 m)(t / s)+(1.1 m)(t /s)2
v(t)  10.1 m/s+(2.2 m/s)(t / s)
Following graph shows position and velocity as a function of time.
• Note: position at minimum where instantaneous
velocity is zero! (Expected from calculus)
So, what can you say about the acceleration?
42
Reading Question 2.1
The slope at a point on a position-versus-time graph
of an object is
A. The object’s speed at that point.
B. The object’s average velocity at that point.
C. The object’s instantaneous velocity at
that point.
D. The object’s acceleration at that point.
E. The distance traveled by the object to
that point.
43
Reading Question 2.1
The slope at a point on a position-versus-time graph
of an object is
A. The object’s speed at that point.
B. The object’s average velocity at that point.
C. The object’s instantaneous velocity at that
point.
D. The object’s acceleration at that point.
E. The distance traveled by the object to
that point.
44
Reading Question 2.2
The area under a velocity-versus-time graph of an
object is
A.
B.
C.
D.
E.
The object’s speed at that point.
The object’s acceleration at that point.
The distance traveled by the object.
The displacement of the object.
This topic was not covered in this chapter.
45
Slide 2-9
Reading Question 2.2
The area under a velocity-versus-time graph of an
object is
A.
B.
C.
D.
E.
The object’s speed at that point.
The object’s acceleration at that point.
The distance traveled by the object.
The displacement of the object.
This topic was not covered in this chapter.
46
Reading Question 2.3
The slope at a point on a velocity-versus-time graph
of an object is
A. The object’s speed at that point.
B. The object’s instantaneous acceleration at that
point.
C. The distance traveled by the object.
D. The displacement of the object.
E. The object’s instantaneous velocity at
that point.
47
Reading Question 2.3
The slope at a point on a velocity-versus-time graph
of an object is
A. The object’s speed at that point.
B. The object’s instantaneous acceleration at that
point.
C. The distance traveled by the object.
D. The displacement of the object.
E. The object’s instantaneous velocity at
that point.
48
Reading Question 2.4
Suppose we define the y-axis to point vertically upward.
When an object is in free fall, it has acceleration in the ydirection
A. ay  g, where g  9.80 m/s2.
B. ay  g, where g  9.80 m/s2.
C. Which is negative and increases in magnitude as
it falls.
D. Which is negative and decreases in magnitude as
it falls.
E. Which depends on the mass of the object.
49
Reading Question 2.4
Suppose we define the y-axis to point vertically upward.
When an object is in free fall, it has acceleration in the ydirection
A. ay  g, where g  9.80 m/s2.
B. ay  g, where g  9.80 m/s2.
C. Which is negative and increases in magnitude as
it falls.
D. Which is negative and decreases in magnitude as
it falls.
E. Which depends on the mass of the object.
50
Reading Question 2.5
At the turning point of an object,
A.
B.
C.
D.
E.
The instantaneous velocity is zero.
The acceleration is zero.
Both A and B are true.
Neither A nor B is true.
This topic was not covered in this chapter.
51
Reading Question 2.5
At the turning point of an object,
A.
B.
C.
D.
E.
The instantaneous velocity is zero.
The acceleration is zero.
Both A and B are true.
Neither A nor B is true.
This topic was not covered in this chapter.
52
QuickCheck 2.3
Here is a motion diagram of a car moving along a straight road:
Which position-versus-time graph matches this motion diagram?
53
QuickCheck 2.3
Here is a motion diagram of a car moving along a straight road:
Which position-versus-time graph matches this motion diagram?
54
QuickCheck 2.4
Here is a motion diagram of a car moving along a straight road:
Which velocity-versus-time graph matches this motion diagram?
E. None of the above.
55
QuickCheck 2.4
Here is a motion diagram of a car moving along a straight road:
Which velocity-versus-time graph matches this motion diagram?
E. None of the above.
56
QuickCheck 2.5
Here is a motion diagram of a car moving along a straight road:
Which velocity-versus-time graph matches this motion diagram?
57
QuickCheck 2.5
Here is a motion diagram of a car moving along a straight road:
Which velocity-versus-time graph matches this motion diagram?
58