1
Physics 200a PSII
1.
Let
A
= 3
i
+ 4
j
and
B
= 5
i

6
j
.
(i) Find
A
+
B
,
A

B
,
2
A
+ 3
B
, and
C
such that
A
+
B
+
C
=
0
.
(ii) Find
A
, the length of
A
and the angle it makes with the xaxis.
A
+
B
= 8
i

2
j
A

B
=

2
i
+ 10
j
2
A
+ 3
B
= 6
i
+ 8
j
+ 15
i

18
j
= 21
i

10
j
C
=

A

B
=

8
i
+ 2
j
(
ii
)
A
=
√
3
2
+ 4
2
= 5
2.
A train is moving with velocity
v
TG
= 3
i
+ 4
j
relative to the ground. A bullet is ﬁred
in the train with velocity
v
BT
= 15
i

6
j
relative to the train. What is the bullets’ velocity
v
BG
relative to the ground?
v
BG
=
v
BT
+
v
= 18
i

2
j
3.
Consider the primed axis rotated relative to the unprimed by an angle
φ
in the coun
terclockwise direction.
(i) Derive the relation
A
x
=
A
0
x
cos
φ

A
0
y
sin
φ
A
y
=
A
0
y
cos
φ
+
A
0
x
sin
φ
that expresses unprimed components in terms of primed components of a vector
~
A
using
class notes if needed to get started.
First note (by drawing a ﬁgure) that the rotated unit vectors are related to the old ones
as follows
i
0
=
i
cos
φ
+
j
sin
φ
j
0
=
j
cos
φ

i
sin
φ
Now we have
A
=
A
0
x
i
0
+
A
0
y
j
0
(1)
=
A
0
x
(
i
cos
φ
+
j
sin
φ
) +
A
0
y
(
j
cos
φ

i
sin
φ
)
(2)
=
i
(
A
0
x
cos
φ

A
0
y
sin
φ
) +
j
(
A
0
y
cos
φ
+
A
0
x
sin
φ
)
(3)
The coeﬃcients of
i
and
j
, are by deﬁnition
A
x
and
A
y
, yielding the desired result.
(ii) Invert these relations to express the primed components in terms of unprimed com
ponents. In doing this remember that the sines and cosines are constants and that we should
treat
A
0
x
and
A
0
y
as unknowns written in terms of knowns
A
x
and
A
y
. (Thus multiply one
equation by something, another by something else, add and subtract etc to isolate the un
knowns. Use simple trig identities)
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To ﬁnd
A
0
x
, we can multiply the equation for
A
x
by cos
φ
, the equation for
A
y
by sin
φ
to get
A
x
cos
φ
=
A
0
x
cos
2
φ

A
0
y
sin
φ
cos
φ
(4)
A
y
sin
φ
=
A
0
y
sin
φ
cos
φ
+
A
0
x
sin
2
φ.
(5)
Adding the two equations gives
A
x
cos
φ
+
A
y
sin
φ
=
A
0
x
(cos
2
φ
+ sin
2
φ
) =
A
0
x
,
because the terms containing
A
0
y
cancel. Similarly, to ﬁnd
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 Fall '08
 RAMAMURTISHANKAR
 Physics, Velocity, Cos, Elementary algebra, ax

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