Problem 2.135
A sports car at point
A
drives down a straight stretch of road
CD
. A police car
at point
B
uses radar to measure the speed of the car, and obtains a reading of
80 km=h
. For the position of the car given below, determine the speed of the car.
Note that the
80 km=h
speed measured by the radar gun is the rate of change of
the distance between the car and the radar gun.
The car is 60% of the distance from point Cto point D.
Solution
Let the actual speed of the car along the line from
C
to
D
be denoted by
v
, and then the vector representation
of this (i.e., the velocity) is
EvDv.ErCD =rCD /
. Using the coordinates of points
C
and
D
given in the problem
statement,
Problem 2.136
Let the road shown be straight between points
C
and
D
and beyond. Determine
the shortest distance between the police car at point
B
and the road. Determine
if the point where the road is closest to the police car falls within segment
CD
,
or outside of this.
Problem 2.137
The manager of a baseball team plans to use a radar gun positioned at point
A
to
measure the speed of pitches for a right-handed pitcher. If the person operating
the radar gun measures a speed
s
when the baseball is one-third the distance
from the release point at
B
to the catcher’s glove at
C
, what is the actual speed
of the pitch? Assume the pitch follows a straight-line path, and express your
answer in terms of
s:
Note that the value
s
measured by the radar gun is the rate
of change of the distance between point Aand the ball.
Solution
Problem 2.138
Repeat Prob. 2.137 for a left-handed pitcher whose release point is D.
Solution
Problem 2.139
Structural member
AB
is to be supported by a strut
CD
. Determine the smallest
length
CD
may have, and specify where
D
must be located for a strut of this
length to be used.
Problem 2.140
Determine the smallest distance between member AB and point E.
Solution
Problem 2.141
In Example 2.13 on p. 76, determine the smallest distance between point
D
and the infinite line passing
through points Aand B. Is this distance the same as the smallest distance to rod AB? Explain.
Problem 2.142
In Example 2.18 on p. 94, determine the smallest distance between point
O
and the infinite line passing
through points Aand B. Is this distance the same as the smallest distance to rod AB? Explain.
Problem 2.143
A building’s roof has “6 in 12” slope in the front and back and “8 in 12” slope on
the sides. Determine the angles
˛
and
ˇ
that should be used for cutting sheets
of plywood so they properly meet along edge
AB
of the roof. Hint: Write the
position vector
ErAB
(where
B
is some point along the edge of the roof) two
ways:
ErAB D ErAC C ErCB
and
ErAB D ErAD C ErDB
. Then use the roof slopes to
help write
ErCB
and
ErDB
such that the magnitudes of the two expressions for
ErAB are the same.
202 Solutions Manual
Problem 2.144
Vectors E
Aand E
Blie in the xy plane.
(a)
Use Eq. (2.48) on p. 101 to evaluate
E
AE
B
, expressing the resulting
vector using Cartesian representation.
(b)
Evaluate
E
AE
B
by computing the determinant of a matrix, using either
Method 1 or Method 2.
Solution
Problem 2.145
Vectors E
Aand E
Blie in the xy plane.
(a)
Use Eq. (2.48) on p. 101 to evaluate
E
AE
B
, expressing the resulting
vector using Cartesian representation.
(b)
Evaluate
E
AE
B
by computing the determinant of a matrix, using either
Method 1 or Method 2.
Solution
Problem 2.146
(a) Evaluate E
AE
B.
(b) Evaluate E
BE
A.
(c) Comment on any differences between the results of Parts (a) and (b).
(d)
Use the dot product to show the result of Part (a) is orthogonal to vectors
E
A
and E
B.
Solution
Problem 2.147
(a) Evaluate E
AE
B.
(b) Evaluate E
BE
A.
(c) Comment on any differences between the results of Parts (a) and (b).
(d)
Use the dot product to show the result of Part (a) is orthogonal to vectors
E
A
and E
B.
Problem 2.148
(a) Evaluate E
AE
B.
(b) Evaluate E
BE
A.
(c) Comment on any differences between the results of Parts (a) and (b).
(d)
Use the dot product to show the result of Part (a) is orthogonal to vectors
E
A
and E
B.
Problem 2.149
(a) Evaluate E
AE
B.
(b) Evaluate E
BE
A.
(c) Comment on any differences between the results of Parts (a) and (b).
(d)
Use the dot product to show the result of Part (a) is orthogonal to vectors
E
A
and E
B.
Problem 2.150
Describe how the cross product operation can be used to determine (or “test”) whether two vectors
E
A
and
E
B
are orthogonal. Is this test as easy to use as the test based on the dot product? Explain, perhaps using an
example to support your remarks. Note: Concept problems are about explanations, not computations.
Solution
Problem 2.151
Imagine a left-hand coordinate system has inadvertently been used for a problem. That is, if the
x
and
y
directions have been selected first, the
´
direction has been taken in the wrong direction for a right-hand
coordinate system. What consequences will this have for dot products and cross products? Perhaps use an
example to support your remarks.
Solution
If a left-hand coordinate system is used, the dot product may still be employed and all remarks made in the
book regarding its use and interpretation are still applicable. To see why, consider
Problem 2.152
The corner of a tent is supported using three ropes having the forces shown. We
wish to compute the sum of the cross products
E
MOD ErOA E
FAB C ErOA
E
FAC C ErOA E
FAD
where
ErOA
is the position vector from points
O
to
A
,
E
FAB
is the force directed from points Ato B, and so on.
(a)
Rather than compute three separate cross products to find
E
MO
, do the
properties of the cross product permit
E
MO
to be found using just one
cross product? Explain.
(b) Determine E
MO.
Solution