Physics For Scientists And Engineers 6E - part 12

S E C T I O N   2 . 7     •     Kinematic Equations Derived from Calculus

45

duration  $t

n

.  From  the  definition  of  average  velocity  we  see  that  the  displacement 

during  any  small  interval,  such  as  the  one  shaded  in  Figure  2.15,  is  given  by

where 

is  the  average  velocity  in  that  interval.  Therefore,  the

displacement  during  this  small  interval  is  simply  the  area  of  the  shaded  rectangle. 
The  total  displacement  for  the  interval  t

f

#

t

i

is  the  sum  of  the  areas  of  all  the

rectangles:

where the symbol + (upper case Greek sigma) signifies a sum over all terms, that is,
over all values of n. In this case, the sum is taken over all the rectangles from t

i

to t

f

.

Now, as the intervals are made smaller and smaller, the number of terms in the sum in-
creases  and  the  sum  approaches  a  value  equal  to  the  area  under  the  velocity–time
graph. Therefore, in the limit n : ,, or $t

n

:

0, the displacement is

(2.14)

or

Note that we have replaced the average velocity 

with the instantaneous velocity v

xn

in the sum. As you can see from Figure 2.15, this approximation is valid in the limit of
very  small  intervals.  Therefore  if  we  know  the  v

x

-t graph  for  motion  along  a  straight

line, we can obtain the displacement during any time interval by measuring the area
under the curve corresponding to that time interval.

The limit of the sum shown in Equation 2.14 is called a 

definite integral and is

written

(2.15)

where v

x

(t) denotes the velocity at any time t. If the explicit functional form of v

x

(t) is

known and the limits are given, then the integral can be evaluated. Sometimes the v

x

-t

graph for a moving particle has a shape much simpler than that shown in Figure 2.15.
For example, suppose a particle moves at a constant velocity v

xi

. In this case, the v

x

-t

graph is a horizontal line, as in Figure 2.16, and the displacement of the particle dur-
ing the time interval $t is simply the area of the shaded rectangle:

As another example, consider a particle moving with a velocity that is proportional to t,
as  in  Figure  2.17.  Taking  v

x

!

a

x

t, where  a

x

is  the  constant  of  proportionality  (the

$

x ! v

xi

 

$

t

   

(when v

x

!

v

xi

!

constant)

lim

$

t

n

:

0

%

n

v

xn

 

$

t

n

!

&

t

f

t

i

v

x

(t)dt

v

xn

Displacement ! area under the v

x

-t graph

$

x ! lim

$

t

n

:

0

%

n

v

xn

$

t

n

$

x !

%

n

v

xn 

$

t

n

v

xn

$

x

n

!

v

xn

 $t

n

v

x

 = v

xi

 = constant

t

 

f

v

xi

t

∆t

t

 

i

v

x

v

xi

Figure 2.16 The velocity–time curve for a

particle moving with constant velocity v

xi

.

The displacement of the particle during the

time interval t

f

#

t

i

is equal to the area of the

shaded rectangle.

Definite integral

acceleration),  we  find  that  the  displacement  of  the  particle  during  the  time  interval
t ! 0 to t ! t

A

is equal to the area of the shaded triangle in Figure 2.17:

Kinematic Equations

We now use the defining equations for acceleration and velocity to derive two of our
kinematic equations, Equations 2.9 and 2.12.

The defining equation for acceleration (Eq. 2.7),

may be written as dv

x

!

a

x

dt or, in terms of an integral (or antiderivative), as

For the special case in which the acceleration is constant, a

x

can be removed from the

integral to give

(2.16)

which is Equation 2.9.

Now let us consider the defining equation for velocity (Eq. 2.5):

We can write this as dx ! v

x

dt, or in integral form as

Because v

x

!

v

xf

!

v

xi

% 

a

x

t, this expression becomes

which is Equation 2.12. 

Besides what you might expect to learn about physics concepts, a very valu-

able skill you should hope to take away from your physics course is the ability to
solve  complicated  problems.  The  way  physicists  approach  complex  situations
and break them down into manageable pieces is extremely useful. On the next
page is a general problem-solving strategy that will help guide you through the
steps. To help you remember the steps of the strategy, they are called Conceptu-
alize, Categorize, Analyze, and Finalize.

 ! v

xi

t %

1

2

a

x

t

2

x

f

#

x

i 

!

&

t

0

(v

xi

%

a

x

t)dt !

&

t

0

v

xi

dt % a

x

&

t

0

tdt ! v

xi

(t # 0) % a

x

"

t

2

2

#

0

#

x

f

#

x

i

!

&

t

0

v

x

dt

v

x

!

dx

dt

v

xf

#

v

xi

!

a

x

&

t

0

dt ! a

x

(t # 0) ! a

x

t

v

xf

#

v

xi

!

&

t

0

a

x

dt

a

x

!

dv

x

dt

$

x !

1

2

(t

A

)(a

x

t

A

) !

1

2

 a

x

t

A

 

 

2

46

C H A P T E R   2     •     Motion in One Dimension

t

v

 

x

 = a

 

x

t

v

 

x

a

 

x

t

A

t

 

A

!

Figure 2.17 The velocity–time curve for a parti-

cle moving with a velocity that is proportional to

the time.

Analyze

•

Now you must analyze the problem and strive for a
mathematical solution. Because you have already cat-
egorized the problem, it should not be too difficult to
select relevant equations that apply to the type of situ-
ation in the problem. For example, if the problem in-
volves a particle moving under constant acceleration,
Equations 2.9 to 2.13 are relevant.

•

Use algebra (and calculus, if necessary) to solve sym-
bolically for the unknown variable in terms of what is
given. Substitute in the appropriate numbers, calcu-
late the result, and round it to the proper number of
significant figures.

Finalize

•

This is the most important part. Examine your nu-
merical answer. Does it have the correct units? Does it
meet your expectations from your conceptualization
of the problem? What about the algebraic form of the
result—before you substituted numerical values?
Does it make sense? Examine the variables in the
problem to see whether the answer would change in a
physically meaningful way if they were drastically in-
creased or decreased or even became zero. Looking
at limiting cases to see whether they yield expected
values is a very useful way to make sure that you are
obtaining reasonable results.

•

Think about how this problem compares with others
you have solved. How was it similar? In what critical
ways did it differ? Why was this problem assigned? You
should have learned something by doing it. Can you
figure out what? If it is a new category of problem, be
sure you understand it so that you can use it as a
model for solving future problems in the same cate-
gory.

When solving complex problems, you may need to
identify a series of sub-problems and apply the problem-
solving strategy to each. For very simple problems, you
probably don’t need this strategy at all. But when you
are looking at a problem and you don’t know what to
do next, remember the steps in the strategy and use them
as a guide.

For practice, it would be useful for you to go back

over the examples in this chapter and identify the Concep-
tualize, Categorize, Analyze, and Finalize steps. In the next
chapter, we will begin to show these steps explicitly in the
examples.

Conceptualize

•

The first thing to do when approaching a problem is
to think about and understand the situation. Study care-
fully any diagrams, graphs, tables, or photographs
that accompany the problem. Imagine a movie, run-
ning in your mind, of what happens in the problem.

•

If a diagram is not provided, you should almost always
make a quick drawing of the situation. Indicate any
known values, perhaps in a table or directly on your
sketch.

•

Now focus on what algebraic or numerical informa-
tion is given in the problem. Carefully read the prob-
lem statement, looking for key phrases such as “starts
from rest” (v

i

!

0), “stops” (v

f

!

0), or “freely falls”

(a

y

! #

g ! # 9.80 m/s

2

).

•

Now focus on the expected result of solving the prob-
lem. Exactly what is the question asking? Will the fi-
nal result be numerical or algebraic? Do you know
what units to expect?

•

Don’t forget to incorporate information from your
own experiences and common sense. What should
a reasonable answer look like? For example, you
wouldn’t expect to calculate the speed of an automo-
bile to be 5 " 10

6

m/s. 

Categorize

•

Once you have a good idea of what the problem is
about, you need to simplify the problem. Remove the
details that are not important to the solution. For
example, model a moving object as a particle. If ap-
propriate, ignore air resistance or friction between a
sliding object and a surface.

•

Once the problem is simplified, it is important to cate-
gorize the problem. Is it a simple plug-in problem, such
that numbers can be simply substituted into a defini-
tion? If so, the problem is likely to be finished when
this substitution is done. If not, you face what we can
call an analysis problem—the situation must be ana-
lyzed more deeply to reach a solution.

•

If it is an analysis problem, it needs to be categorized
further. Have you seen this type of problem before?
Does it fall into the growing list of types of problems
that you have solved previously? Being able to classify
a problem can make it much easier to lay out a plan
to solve it. For example, if your simplification shows
that the problem can be treated as a particle moving
under constant acceleration and you have already
solved such a problem (such as the examples in Sec-
tion 2.5), the solution to the present problem follows
a similar pattern.

G E N E RAL P R O B L E M -S O LVI N G STRATEGY

47

48

C H A P T E R   2     •     Motion in One Dimension

After a particle moves along the x axis from some initial position x

i

to some final posi-

tion x

f

, its 

displacement is

(2.1)

The 

average velocity of a particle during some time interval is the displacement

$

x divided by the time interval $t during which that displacement occurs:

(2.2)

The 

average speed of a particle is equal to the ratio of the total distance it travels

to the total time interval during which it travels that distance:

(2.3)

The 

instantaneous velocity of a particle is defined as the limit of the ratio $x/$t

as $t approaches zero. By definition, this limit equals the derivative of x with respect to
t, or the time rate of change of the position:

(2.5)

The 

instantaneous speed of a particle is equal to the magnitude of its instantaneous

velocity.

The 

average acceleration of a particle is defined as the ratio of the change in its

velocity $v

x

divided by the time interval $t during which that change occurs:

(2.6)

The 

instantaneous acceleration is equal to the limit of the ratio $v

x

/$t as $t ap-

proaches 0. By definition, this limit equals the derivative of v

x

with respect to t, or the

time rate of change of the velocity:

(2.7)

When  the  object’s  velocity  and  acceleration  are  in  the  same  direction,  the  object  is
speeding up. On the other hand, when the object’s velocity and acceleration are in op-
posite directions, the object is slowing down. Remembering that F a is a useful way to
identify the direction of the acceleration.

The 

equations of kinematics for a particle moving along the x axis with uniform

acceleration a

x

(constant in magnitude and direction) are

(2.9)

(2.11)

(2.12)

(2.13)

An object falling freely in the presence of the Earth’s gravity experiences a free-fall

acceleration  directed  toward  the  center  of  the  Earth.  If  air  resistance  is  neglected,  if
the  motion  occurs  near  the  surface  of  the  Earth,  and  if  the  range  of  the  motion  is
small  compared  with  the  Earth’s  radius,  then  the  free-fall  acceleration  g is  constant
over the range of motion, where g is equal to 9.80 m/s

2

.

Complicated  problems  are  best  approached  in  an  organized  manner.  You  should

be able to recall and apply the Conceptualize, Categorize, Analyze, and Finalize steps of the
General Problem-Solving Strategy when you need them.

v

xf

2

!

v

xi

2

%

2a

x

(x

f

#

x

i

)

x

f

!

x

i

%

v

xi

t %

1

2

a

x

t

2

x

f

!

x

i

%

v

x

t ! x

i

%

1

2

(v

xi

%

v

xf

)t

v

xf 

!

v

xi

%

a

x

t

)

a

x 

!  lim

$

t : 0

$

v

x

$

t

!

dv

x

dt

a

x 

! 

$

v

x

$

t

!

v

xf

#

v

xi

t

f

#

t

i

v

x 

!  lim

$

t : 0

$

x

$

t

!

dx

dt

Average speed !

total distance

total time

v

x 

! 

$

x

$

t

$

x 

! x

f

#

x

i

S U M M A R Y

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