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*22. The velocity of an aircraft | a + b | d" | a| + |b |.
is 200 km/hr due west. A northwest 31. An airplane flies due east at
wind of 50 km/hr is blowing. 200 km/hr straight from city A to
(a) What is the velocity of the city B a distance of 200 km. A wind
aircraft relative to the ground? of 40 km/hr from the northwest is
(b) If the pilot s destination is due blowing. If the pilot doesn t make
west, at what angle should he point any corrections, where will the
his plane to get there? (c) If his plane be in 1 hr?
destination is 400 km due west, 32. Given vectors a and b,
how long will it take him to get
where a = 50, ¸ = 330, b = 80,
1
there?
and ¸ = 1280, find (a) a + b, (b) a -
2
23. A plane flies east for 50.0
b, (c) a - 2b, (d) 3a + b, (e) 2a - b,
km, then at an angle of 30.00 north
and (f) 2b - a.
of east for 75.0 km. In what
33. In the accompanying figure
direction should it now fly and how
the tension T in the cable is 200 N.
far, such that it will be 200 km
Find the vertical component T and
y
northwest of its original position?
the horizontal component T of this
x
Diagram for problem 26.
*24. The current in a river flows
tension.
south at 7 km/hr. A boat starts
27. Find the resultant of the
straight across the river at 19
following three forces. The
km/hr relative to the water.
magnitudes of the forces are
(a) What is the speed of the boat
F = 2.00 N, F = 8.00 N, and
1 2
relative to the land? (b) If the river
F = 6.00 N.
3
is 1.5 km wide, how long does it
take the boat to cross the river?
(c) If the boat sets out straight for
the opposite side, how far south will
it reach the opposite shore? (d) If we
want to have the boat go straight
across the river, at what angle
should the boat be headed?
*25. Show that if the angle
between vectors a and b is an acute
angle, then the sum a + b becomes
Diagram for problem 33.
the main diagonal of the
parallelogram and the difference a
*34. In the accompanying
- b becomes the minor diagonal of
diagram w is 5 N and w is 3 N.
1 2
the parallelogram. Also show that if
Find the angle ¸ such that the
the angle is obtuse the results are
component of w parallel to the
1
reversed.
Diagram for problem 27.
incline is equal to w .
2
26. Find the resultant of the
following three vectors. The
*28. Show that for three
magnitudes of the vectors are
nonparallel vectors all in the same
a = 5.00 km, b = 10.0 km, and
plane, any one of them can be
c = 20.0 km.
represented as a linear sum of the
other two.
*29. A unit vector is a vector
that has a magnitude of one unit
and is in a specified direction. If a
unit vector i is defined to be in the
x-direction, and a unit vector j is
defined to be in the y-direction,
show that any vector a can be
Diagram for problem 34.
written in the form
*35. In the accompanying
a = a i + a j
x y
diagram w = 2 N, w = 5 N, and ¸ =
1 2
Chapter 2 Vectors 2-17
650. Find the angle Æ such that the
components of the two forces
parallel to the inclines are equal.
Diagram for problem 36.
Diagram for problem 37.
*37. In projectile motion in two
Diagram for problem 35.
dimensions the projectile is located
Interactive Tutorials
by the displacement vector r at the
1
*36. In the accompanying
38. The components of a vector.
time t and by the displacement
1
diagram w = 50 N, and ¸ = 100.
A 50.0-N force is directed at an
vector r at t , as shown in the
2 2
What must be the value of F such
angle of 500 above the horizontal.
diagram. If r = 20 m, ¸ = 600, r =
1 1 2
that w will be held in place? What
Resolve this force into vertical and
25 m, and ¸ = 250, find the
2
happens if the angle is doubled to
horizontal components.
magnitude and direction of the
200?
39. Resultant vector. Find the
vector r - r .
2 1
resultant of any number of force
vectors (up to five vectors).
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2-18 Mechanics
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