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Transcript
Chapter 3 | Two-Dimensional Kinematics
Problems & Exercises
3.2 Vector Addition and Subtraction: Graphical
Methods
Use graphical methods to solve these problems. You may
assume data taken from graphs is accurate to three
digits.
1. Find the following for path A in Figure 3.54: (a) the total
distance traveled, and (b) the magnitude and direction of the
displacement from start to finish.
Figure 3.56
6. Repeat the problem above, but reverse the order of the two
legs of the walk; show that you get the same final result. That
is, you first walk leg B , which is 20.0 m in a direction exactly
40º south of west, and then leg A , which is 12.0 m in a
20º west of north. (This problem shows that
A + B = B + A .)
direction exactly
Figure 3.54 The various lines represent paths taken by different people
walking in a city. All blocks are 120 m on a side.
2. Find the following for path B in Figure 3.54: (a) the total
distance traveled, and (b) the magnitude and direction of the
displacement from start to finish.
3. Find the north and east components of the displacement
for the hikers shown in Figure 3.52.
4. Suppose you walk 18.0 m straight west and then 25.0 m
straight north. How far are you from your starting point, and
what is the compass direction of a line connecting your
starting point to your final position? (If you represent the two
legs of the walk as vector displacements A and B , as in
Figure 3.55, then this problem asks you to find their sum
R = A + B .)
7. (a) Repeat the problem two problems prior, but for the
second leg you walk 20.0 m in a direction 40.0º north of east
(which is equivalent to subtracting
B from A —that is, to
finding
R′ = A − B ). (b) Repeat the problem two problems
40.0º south
of west and then 12.0 m in a direction 20.0º east of south
(which is equivalent to subtracting A from B —that is, to
finding R′′ = B - A = - R′ ). Show that this is the case.
prior, but now you first walk 20.0 m in a direction
8. Show that the order of addition of three vectors does not
affect their sum. Show this property by choosing any three
vectors A , B , and C , all having different lengths and
directions. Find the sum
A + B + C then find their sum
when added in a different order and show the result is the
same. (There are five other orders in which A , B , and C
can be added; choose only one.)
9. Show that the sum of the vectors discussed in Example
3.2 gives the result shown in Figure 3.24.
10. Find the magnitudes of velocities
v A and v B in Figure
3.57
Figure 3.55 The two displacements
displacement
R
having magnitude
A
R
and
B
add to give a total
and direction
5. Suppose you first walk 12.0 m in a direction
θ.
20º west of
north and then 20.0 m in a direction 40.0º south of west.
How far are you from your starting point, and what is the
compass direction of a line connecting your starting point to
your final position? (If you represent the two legs of the walk
as vector displacements A and B , as in Figure 3.56, then
this problem finds their sum
R = A + B .)
Figure 3.57 The two velocities
vA
11. Find the components of
and
vB
add to give a total
v tot .
v tot along the x- and y-axes in
Figure 3.57.
18