Chapter 2
Kinematics: Description of Motion
A bit of terminology…
•
•
Kinematics (the topic
of the next two
chapters) deals only
with the description
of motion, without
considering its
causes. It involves
concepts of position,
displacement,
speed, velocity, and
acceleration.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Dynamics (the rest of
PC141) concerns the
causes of motion. It
begins with the
subject of force, which
leads to the concepts
of mechanical energy,
linear momentum,
and angular
momentum. These
are then linked back to
the kinematic
concepts in order to
fully describe the
motion.
Slide 1
2.1 Distance and Speed
All motion involves changing an object’s position. In studying
kinematics, we wish to analyze (or predict) where an object is (or
will be) at various times.
The simplest description of motion is the distance that an object
travels, which is the total path length traversed in moving
between two points. Distance depends not only on the position
of the two points, but on the particular path taken during the
trip.
Distance is a scalar quantity (it has a
magnitude – and units – but no
direction).
The text uses the symbol d to
represent distance.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 2
2.1 Distance and Speed
When an object is in motion, it moves a certain position in a
given amount of time. The relationship between these two
parameters tells us the object’s speed.
Average speed 𝑠 is the distance traveled divided by the time
elapsed during the trip. Note that the bar over s is used to
denote an average quantity, not to indicate a vector.
We denote the start and end times of the trip as t1 and t2,
respectively. The time elapsed is notated as Δt =t2 – t1 (“Δ” is the
Greek capital letter “delta”, which is used in the sciences to
represent a “difference” or “change”). Then,
𝑑
𝑑
𝑠=
=
∆𝑡 𝑡2 − 𝑡1
As the ratio of a distance and a time, speed has SI units of m/s.
Other common units are km/h and miles/h (often written mph).
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 3
2.1 Distance and Speed
We can also define an instantaneous speed s as the speed at a
particular moment in time. That is, it is the average speed in the
limit as Δt approaches zero. The speedometer in a car shows
instantaneous speed (more or less…all sensors have a finite
“sensing time”, but it’s very short).
Instantaneous speed is very important in calculus-based physics
courses, but we won’t use it very often in PC141.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 4
2.2 1D Displacement and Velocity
For the remainder of this chapter, we will only consider motion
in one dimension – that is, objects are constrained to move back
and forth along a straight line. This may seem like a very
restrictive view of the world, but it will turn out to be quite
useful.
ORIGIN
To begin, we will label the
straight line as the x-axis.
The position of an object is a
function of time, labeled x(t).
Its value depends on where
we place the origin, where x
= 0. This location is entirely
up to us.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 5
2.2 1D Displacement and Velocity
Displacement is the straight-line distance between two points,
along with an indication of the direction (which is positive if x
increases from the first point to the second, and negative if x
decreases). The man in the figure has a displacement of +8 m if
he walks from x1 to x2, where x1 = x(t1) and x2 = x(t2).
Mathematically, we write
∆𝑥 = 𝑥2 − 𝑥1
Since it has both a
magnitude and a direction,
displacement is a vector
quantity.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 6
2.2 1D Displacement and Velocity
Velocity tells us both how fast an object is moving and in what
direction.
The average velocity 𝒗 between time t1 (when the object is at
position x1) and time t2 (when the object is at position x2) is
displacement
∆𝑥 𝑥2 − 𝑥1
𝑣=
=
=
total travel time ∆𝑡
𝑡2 − 𝑡1
Average velocity and average speed are not the same thing. For
example, if you take a walk to the corner store and back, you
might travel a total of 300 m in 5 minutes. Your speed is
(300 m) / (300 s) = 1 m/s. However, since your starting position
and ending position are the same, your displacement is Δx = 0, so
your average velocity is zero.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 7
2.2 Displacement and Velocity
As with speed, we can also define an
instantaneous velocity v as the
velocity at a particular moment in
time. That is, it is the average
velocity in the limit as Δt approaches
zero:
∆𝑥
𝑣 = lim
∆𝑡→0
∆𝑡
Uniform motion is motion in which
the velocity is constant (in both
magnitude and direction). In
uniform motion, the average
velocity and instantaneous velocity
are identical.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 8
2.2 Displacement and Velocity
Graphical analysis is often used in PC141 (and in most science
courses). For 1D motion, we can plot position as a function of
time, x vs. t. Then, the description of average velocity from slide
∆𝑥
7 indicates that it is equal to the slope of this graph, .
∆𝑡
Thus, we see that a positive average velocity produces a graph
with positive slope (slide 8), and a negative average velocity
produces a graph with negative slope (below).
If the velocity is constant,
then the slope can never
change. In this case, the
graph is a straight line.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 9
2.2 Displacement and Velocity
If the velocity is not constant, then the graph is a curve. The plot
below shows position vs. time for an object that speeds up,
slows down, reverses direction, etc. We will discuss it in class.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 10
2.3 Acceleration
When an object’s velocity is not constant, we say that the object
has a non-zero acceleration. The relationship between
acceleration and velocity is identical to that between velocity
and position. That is, acceleration is the time rate of change of
velocity. The average acceleration is defined as
change in velocity
∆𝑣 𝑣2 − 𝑣1
𝑎=
=
=
time over which velocity changes ∆𝑡
𝑡2 − 𝑡1
SI units for acceleration are m/s2. Don’t waste too much time
contemplating the concept of a “squared second”… the units
merely refer to the fact that the ratio of a velocity to a time is
measured as (m/s) / s.
We can also express the instantaneous acceleration as
∆𝑣
𝑎 = lim
∆𝑡→0 ∆𝑡
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 11
2.3 Acceleration
Since velocity is a vector, so is acceleration. A change in velocity
might indicate a change in speed, or a change in direction, or
both. Either of these will produce a non-zero acceleration.
The case of a change in speed (not direction) is shown below.
The directions of velocity and acceleration and their relation to
whether an object is “speeding up” or “slowing down” are rather
confusing at first.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 12
2.3 Acceleration
In the figure, the positive
direction (the direction of
increasing x) is to the right.
Since Δt is positive (it’s the
difference between a later
time and an earlier time),
the sign of 𝑎 is identical to
the sign of ∆𝑣.
We will discuss this figure in
class.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 13
2.3 Acceleration
The case of a change in direction (not speed) is shown below on
the left. We won’t analyze this one quite yet – since this motion
takes place in 2 dimensions, we will save it for the next chapter.
It is also possible to change both speed and direction, as shown
on the right.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 14
2.3 Acceleration
In general, acceleration can be a function of time, a(t). However,
when acceleration is constant, we can derive many simple
relationships among acceleration, velocity, and position. The
text implies that this case is examined “for simplicity”… I
disagree! There are many examples of physical situations for
which acceleration really is constant. In fact, much of PC141 is
concerned with these situations.
To begin, we need to adjust our notation a bit. Until now, we
assumed that there were two specific times, t1 and t2, at which
an object had position x1 and x2 and velocity v1 and v2. Here, we
will change the initial conditions to t0 = 0, x0, and v0, and consider
the final conditions as variables t, x, and v. Note that we can set
the initial time to zero since only changes in time have any
physical meaning.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 15
2.3 Acceleration
Next, we note that if acceleration is constant, then the average
acceleration over a duration of time and the instantaneous
acceleration at any particular time must be equal: 𝑎 = 𝑎.
Rearranging our original equation for average acceleration, we
have
𝑣2 − 𝑣1 𝑣 − 𝑣0
𝑎=𝑎=
𝑡2 − 𝑡1
=
𝑡−0
Then, rearranging once more, we find that
𝑣 = 𝑣0 + 𝑎𝑡
In other words, when acceleration is constant, the velocity at
any future time t is equal to the velocity at the initial time plus
the product of acceleration and time.
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 16
2.3 Acceleration
We see from the previous equation that the function 𝑣(𝑡)is a
straight line, with slope 𝑎 and intercept 𝑣0 .
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 17
Problem #1: Graphical Analysis
WBL LP 2.6
An object has a constant, non-zero acceleration. A graph
of position vs. time for this object is:
PC141 Intersession 2013
A
A horizontal line
B
A nonhorizontal and nonvertical straight line
C
A vertical line
D
A curve
Day 3 – May 9 – WBL 2.1-2.3
Slide 18
Problem #2: Deceleration
WBL LP 2.11
Which of the following is true for a deceleration?
A
The velocity remains constant
B
The acceleration is negative
C
The acceleration is in the direction opposite to the velocity
D
The acceleration is zero
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 19
Problem #3: Average Speed
Your drive 4.00 km at 50.0 km/h and then 4.00 km at 100
km/h. Your average speed for the entire 8.00 km trip is…
A
Less than 75.0 km/h
B
Equal to 75.0 km/h
C
Greater than 75.0 km/h
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 20
Problem #4: Nerve Conduction
The human body contains different types of nerves, and the speed at which
impulses travel along these nerves strongly depends on the particular type. The
sensation of touch relies on Aα receptors, which conduct impulses at roughly 100
m/s. The sensation of pain relies on C receptors, which conduct impulses at
about 0.6 m/s.
Assume that your left big toe is 160 cm from your brain. What is the time lag
between realizing that you’ve stubbed your toe and feeling the resulting pain?
Solution: In class
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 21
Problem #5: Train Trip
WBL EX 2.7
A train makes a round trip on a straight, level track. The first half of the trip is
300 km, and is traveled at a speed of 75 km/h. After a 0.50-hour layover, the
train returns to its original location at a speed of 85 km/h. What is the train’s (a)
average speed, and (b) average velocity?
Solution: In class
PC141 Intersession 2013
Day 3 – May 9 – WBL 2.1-2.3
Slide 22