65
4-2 AVE RAG E VE LOCITY AN D I NSTANTAN EOUS VE LOCITY
Average Velocity and Instantaneous Velocity
If a particle moves from one point to another, we might need to know how fast it
moves. Just as in Chapter 2, we can define two quantities that deal with “how
fast”: average velocity and instantaneous velocity. However, here we must consider these quantities as vectors and use vector notation.
If a particle moves through a displacement
in a time interval t, then its
average velocity
is
or
(4-8)
This tells us that the direction of
(the vector on the left side of Eq. 4-8) must
be the same as that of the displacement
(the vector on the right side). Using
Eq. 4-4, we can write Eq. 4-8 in vector components as
(4-9)
For example, if a particle moves through displacement
in
2.0 s, then its average velocity during that move is
That is, the average velocity (a vector quantity) has a component of 6.0 m/s along
the x axis and a component of 1.5 m/s along the z axis.
When we speak of the velocity of a particle, we usually mean the particle’s
instantaneous velocity at some instant. This is the value that
approaches
in the limit as we shrink the time interval t to 0 about that instant. Using the language of calculus, we may write as the derivative
(4-10)
Figure 4-3 shows the path of a particle that is restricted to the xy plane. As
the particle travels to the right along the curve, its position vector sweeps to the
right. During time interval t, the position vector changes from
to
and the
particle’s displacement is
.
To find the instantaneous velocity of the particle at, say, instant t 1 (when the
particle is at position 1), we shrink interval t to 0 about t 1 . Three things happen
as we do so. (1) Position vector in Fig. 4-3 moves toward so that
shrinks
⌬ r
:
r
:
1
r
:
2
⌬
⌬ r
:
r
:
2
r
:
1
⌬
v
: ϭ
d r
:
dt
.
v
:
⌬
v
:
avg
v
:
v
:
v
:
avg ϭ
⌬ r
:
⌬t
ϭ
(12 m)i ˆ ϩ (3.0 m)k ˆ
2.0 s
ϭ (6.0 m/s)i ˆ ϩ (1.5 m/s)k ˆ .
(12 m)i ˆ ϩ (3.0 m)k ˆ
v
:
avg ϭ
⌬xi ˆ ϩ ⌬yj ˆ ϩ ⌬zk ˆ
⌬t
ϭ
⌬x
⌬t
i ˆ ϩ
⌬y
⌬t
j ˆ ϩ
⌬z
⌬t
k ˆ .
⌬ r
:
v
:
avg
v
:
avg ϭ
⌬ r
:
⌬t
.
average velocity ϭ
displacement
time interval
,
v
:
avg
⌬
⌬ r
:
Figure 4-3 The displacement
of a particle
during a time interval , from position 1 with
position vector at time t 1 to position 2
with position vector at time t 2 . The tangent
to the particle’s path at position 1 is shown.
r
:
2
r
:
1
⌬t
⌬ r
:
r 1
r 2
Path
Tangent
O
y
x
1
2
r
∆
As the particle moves,
the position vector
must change.
This is the
displacement.
4-2 AVE RAG E VE LOCITY AN D I NSTANTAN EOUS VE LOCITY
Average Velocity and Instantaneous Velocity
If a particle moves from one point to another, we might need to know how fast it
moves. Just as in Chapter 2, we can define two quantities that deal with “how
fast”: average velocity and instantaneous velocity. However, here we must consider these quantities as vectors and use vector notation.
If a particle moves through a displacement
in a time interval t, then its
average velocity
is
or
(4-8)
This tells us that the direction of
(the vector on the left side of Eq. 4-8) must
be the same as that of the displacement
(the vector on the right side). Using
Eq. 4-4, we can write Eq. 4-8 in vector components as
(4-9)
For example, if a particle moves through displacement
in
2.0 s, then its average velocity during that move is
That is, the average velocity (a vector quantity) has a component of 6.0 m/s along
the x axis and a component of 1.5 m/s along the z axis.
When we speak of the velocity of a particle, we usually mean the particle’s
instantaneous velocity at some instant. This is the value that
approaches
in the limit as we shrink the time interval t to 0 about that instant. Using the language of calculus, we may write as the derivative
(4-10)
Figure 4-3 shows the path of a particle that is restricted to the xy plane. As
the particle travels to the right along the curve, its position vector sweeps to the
right. During time interval t, the position vector changes from
to
and the
particle’s displacement is
.
To find the instantaneous velocity of the particle at, say, instant t 1 (when the
particle is at position 1), we shrink interval t to 0 about t 1 . Three things happen
as we do so. (1) Position vector in Fig. 4-3 moves toward so that
shrinks
⌬ r
:
r
:
1
r
:
2
⌬
⌬ r
:
r
:
2
r
:
1
⌬
v
: ϭ
d r
:
dt
.
v
:
⌬
v
:
avg
v
:
v
:
v
:
avg ϭ
⌬ r
:
⌬t
ϭ
(12 m)i ˆ ϩ (3.0 m)k ˆ
2.0 s
ϭ (6.0 m/s)i ˆ ϩ (1.5 m/s)k ˆ .
(12 m)i ˆ ϩ (3.0 m)k ˆ
v
:
avg ϭ
⌬xi ˆ ϩ ⌬yj ˆ ϩ ⌬zk ˆ
⌬t
ϭ
⌬x
⌬t
i ˆ ϩ
⌬y
⌬t
j ˆ ϩ
⌬z
⌬t
k ˆ .
⌬ r
:
v
:
avg
v
:
avg ϭ
⌬ r
:
⌬t
.
average velocity ϭ
displacement
time interval
,
v
:
avg
⌬
⌬ r
:
Figure 4-3 The displacement
of a particle
during a time interval , from position 1 with
position vector at time t 1 to position 2
with position vector at time t 2 . The tangent
to the particle’s path at position 1 is shown.
r
:
2
r
:
1
⌬t
⌬ r
:
r 1
r 2
Path
Tangent
O
y
x
1
2
r
∆
As the particle moves,
the position vector
must change.
This is the
displacement.