Physics For Scientists And Engineers 6E - part 28
Problems
109
68.
In a television picture tube (a cathode ray tube) electrons
are emitted with velocity v
i
from a source at the origin of
coordinates. The initial velocities of different electrons
make different angles ' with the x axis. As they move a
distance D along the x axis, the electrons are acted on by
a constant electric field, giving each a constant accelera-
tion a in the x direction. At x " D the electrons pass
through a circular aperture, oriented perpendicular to the
x axis. At the aperture, the velocity imparted to the elec-
trons by the electric field is much larger than v
i
in magni-
tude. Show that velocities of the electrons going through
the aperture radiate from a certain point on the x axis,
which is not the origin. Determine the location of this
point. This point is called a virtual source, and it is impor-
tant in determining where the electron beam hits the
screen of the tube.
69.
A fisherman sets out upstream from Metaline Falls on the
Pend Oreille River in northwestern Washington State. His
small boat, powered by an outboard motor, travels at a
constant speed v in still water. The water flows at a lower
constant speed v
w
. He has traveled upstream for 2.00 km
when his ice chest falls out of the boat. He notices that
the chest is missing only after he has gone upstream for
another 15.0 minutes. At that point he turns around and
heads back downstream, all the time traveling at the same
speed relative to the water. He catches up with the float-
ing ice chest just as it is about to go over the falls at his
starting point. How fast is the river flowing? Solve this
problem in two ways. (a) First, use the Earth as a refer-
ence frame. With respect to the Earth, the boat travels up-
stream at speed v # v
w
and downstream at v & v
w
. (b) A
second much simpler and more elegant solution is ob-
tained by using the water as the reference frame. This ap-
proach has important applications in many more compli-
cated problems; examples are calculating the motion of
rockets and satellites and analyzing the scattering of sub-
atomic particles from massive targets.
70.
The water in a river flows uniformly at a constant speed
of 2.50 m/s between parallel banks 80.0 m apart. You are
to deliver a package directly across the river, but you can
swim only at 1.50 m/s. (a) If you choose to minimize the
time you spend in the water, in what direction should
you head? (b) How far downstream will you be carried?
(c) What If? If you choose to minimize the distance
downstream that the river carries you, in what direction
should you head? (d) How far downstream will you be
carried?
71.
An enemy ship is on the east side of a mountain island,
as shown in Figure P4.71. The enemy ship has maneuvered
to within 2 500 m of the 1 800-m-high mountain peak
and can shoot projectiles with an initial speed of
250 m/s. If the western shoreline is horizontally 300 m
from the peak, what are the distances from the western
shore at which a ship can be safe from the bombardment
of the enemy ship?
72.
In the What If? section of Example 4.7, it was claimed that
the maximum range of a ski-jumper occurs for a launch
angle ' given by
where . is the angle that the hill makes with the horizontal
in Figure 4.16. Prove this claim by deriving the equation
above.
Answers to Quick Quizzes
4.1 (b). An object moving with constant velocity has
!
v " 0, so, according to the definition of acceleration,
a " !v/!t " 0. Choice (a) is not correct because a parti-
cle can move at a constant speed and change direction.
This possibility also makes (c) an incorrect choice.
4.2 (a). Because acceleration occurs whenever the velocity
changes in any way—with an increase or decrease in
' "
45) #
.
2
10.0 m/s
15.0
°
50.0
°
Figure P4.67
2 500 m
300 m
1 800 m
v
i
v
i
= 250 m/s
θ
H
θ
θ
L
θ
Figure P4.71