Environmental Engineering Reference
In-Depth Information
PROBLEMS 1
1.1 For the vectors
a
=
i
+
j
−
2
k
and
b
=
3
i
−
j
+
k
, find:
(a) the vectors
c
=
a
+
b
and
d
=
a
−
b
;
(b) the magnitudes of
a
,
b
and
c
;
(c) a unit vector in the same direction as
a
.
1.2 A bird leaves its nest and flies 100 m NE (i.e. at a bearing of 45
◦
) then 150 m
at a bearing of 150
◦
and finally 50 m due W, where it lands in a tree. How
far is the tree from the nest? In what direction is the tree from the nest?
1.3 The vertices of a triangle, A, B and C, have position vectors
r
A
,
r
B
and
r
C
.
Write down
−
AB
, the position of B relative to A, in terms of
r
A
and
r
B
. Hence
show that the vector sum of the successive sides of the triangle (
−
AB
+
−
BC
+
−
CA
) is zero. Draw a sketch to demonstrate this result geometrically.
1.4 Hubble found that distant galaxies are receding from us with a speed pro-
portional to their distance from the Earth. The velocity of the
i
-th galaxy is
given by
v
i
=
H
0
r
i
,
where
r
i
is the position vector of that galaxy with respect to us at the origin,
and
H
0
is a constant (known as Hubble's constant). Show that this recession
of the galaxies does
not
imply that we are at the centre of the Universe.
1.5 For the vectors
A
=
i
+
j
+
k
,
B
=
2
i
−
2
j
−
2
k
,
C
=
4
i
−
j
−
3
k
,
D
=−
i
+
j
+
k
,
find their magnitudes and the scalar products
A
D
.
Hence find the angles between
A
and each of
B
,
C
and
D
and that between
B
and
D
. Evaluate the vector products
A
·
B
,
A
·
C
,
A
·
D
and
B
·
D
. Check that
the magnitudes of these agree with the corresponding geometrical expressions
(
×
B
,
A
×
D
and
B
×
sin
θ
etc.).
1.6 A charged particle is accelerated uniformly from rest in an electric field. If
after 1.0 nanoseconds the particle has travelled 10
|
A
×
B
|=|
A
||
B
|
µ
m, work out its acceler-
ation.
1.7 A coin is dropped from the top of a tall building. If an observer on the ground
measures the speed of the coin immediately before impact to be 65.0 ms
−
1
,
how tall is the building? For how long was the coin falling? Neglect effects
due to air resistance.
1.8 A missile malfunctions in flight and has a subsequent trajectory described by
the position vector (
s
) at time (
t
), given by,
0
.
005
t
2
k
,
=
+
−
s
0
.
3
t
i
0
.
5
t
j
