316
Chapter 3
EVALUATE:
To travel due west the velocity of the plane relative to the air must have a westward component and
also a component that is northward, opposite to the wind direction.
Figure 3.40
3.41.
IDENTIFY:
Relative velocity problem in two dimensions. His motion relative to the earth (time displacement)
depends on his velocity relative to the earth so we must solve for this velocity.
(a) SET UP:
View the motion from above.
The velocity vectors in the problem are:
M/E
,
v
!
the velocity of the man relative to the earth
W/E
,
v
!
the velocity of the water relative to the earth
M/W
,
v
!
the velocity of the man relative to the water
The rule for adding these velocities is
M/E
M/W
W/E
v=
v +
v
!
!!
Figure 3.41a
The problem tells us that
W/E
v
!
has magnitude 2.0 m/s and direction due south. It also tells us that
M/W
v
!
has
magnitude 4.2 m/s and direction due east. The vector addition diagram is then as shown in Figure 3.41b
This diagram shows the vector addition
M/E
M/W
W/E
v
!
and also has
M/W
v
!
and
W/E
v
!
in their
specified directions. Note that the
vector diagram forms a right triangle.
Figure 3.41b
The Pythagorean theorem applied to the vector addition diagram gives
22 2
M/E
M/W
W/E
.
vv v
=+
EXECUTE:
22
2
2
M/E
M/W
W/E
(4.2 m/s)
(2.0 m/s)
4.7 m/s
vv
v
=
+
=
M/W
W/E
4.2 m/s
tan
2.10;
2.0 m/s
v
v
θ
==
=
65 ;
=
°
or
90
25 .
φ
=°
−=°
The velocity of the man relative to the earth has magnitude 4.7 m/s and direction 25 S
°
of E.
(b)
This requires careful thought. To cross the river the man must travel 800 m due east relative to the earth. The
man&s velocity relative to the earth is
M/E
.
v
!
But, from the vector addition diagram the eastward component of
M/E
v
equals
M/W
4.2 m/s.
v
=
Thus
0
800 m
190 s.
4.2 m/s
x
xx
t
v
−
=
(c)
The southward component of
M/E
v
!
equals
W/E
2.0 m/s.
v
=
Therefore, in the 190 s it takes him to cross the river
the distance south the man travels relative to the earth is
0
(2.0 m/s)(190 s)
380 m.
y
yy v
t
−= =
=
EVALUATE:
If there were no current he would cross in the same time, (800 m)/(4.2 m/s) 190 s.
=
The current
carries him downstream but doesn&t affect his motion in the perpendicular direction, from bank to bank.
3.42.
IDENTIFY:
Use the relation that relates the relative velocities.
SET UP:
The relative velocities are the water relative to the earth,
W/E
v
!
, the boat relative to the water,
B/W
v
!
, and
the boat relative to the earth,
B/E
v
!
.
B/E
v
!
is due east,
W/E
v
!
is due south and has magnitude 2.0 m/s.
B/W
4.2 m/s
v
=
.
B/E
B/W
W/E
!! !
. The velocity addition diagram is given in Figure 3.42.
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317
EXECUTE:
(a)
Find the direction of
B/W
v
!
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 Spring '06
 Buchler
 Physics, Velocity, m/s, tan φ

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