Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 3

Kinematics in Two or Three Dimensions; Vectors - all with Video Answers

Educators

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Chapter Questions

Problem 1

(1) A car is driven 225 $\mathrm{km}$ west and then 78 $\mathrm{km}$ southwest $\left(45^{\circ}\right) .$
What is the displacement of the car from the point of origin
(magnitude and dircction)? Draw a diagram.

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

(1) A delivery truck travels 28 blocks north, 16 blocks east, and 26 blocks south. What is its final displacement from the origin? Assume the blocks are equal length.

Zachary Warner

Numerade Educator

Problem 3

$$ \begin{array}{l}{\text { (1) If } V_{x}=7.80 \text { units and } V_{y}=-6.40 \text { units, determine the }} \\ {\text { magnitude and direction of } \vec{\mathbf{v}} .}\end{array} $$

Averell Hause

Carnegie Mellon University

Problem 4

(II) Graphically determine the resultant of the following three vector displacements:
(1) $24 \mathrm{~m}, 36^{\circ}$ north of east;
(2) $18 \mathrm{~m}$ $37^{\circ}$ east of north; and
(3) $26 \mathrm{~m}, 33^{\circ}$ west of south.

Zachary Warner

Numerade Educator

Problem 5

(II) $\hat{\mathbf{v}}$ is a vector 24.8 units in magnitude and points at an angle of $23.4^{\circ}$ above the negative $x$ axis. $(a)$ Sketch this vector. (b) Calculate $V_{x}$ and $V_{y}-(c)$ Use $V_{x}$ and $V_{y}$ .to obtain (again) the magnitude and direction of $\vec{\mathbf{v}}$[Note. Part $(c)$ is a good way to check if you've resolved your vector correctly.

Averell Hause

Carnegie Mellon University

Problem 6

$$\begin{array}{l}{\text { (II) Figure } 36 \text { shows two vectors, } \vec{\mathbf{A}} \text { and } \vec{\mathbf{B}}, \text { whose magni- }} \\ {\text { tudes are } A=6.8 \text { units and } B=5.5 \text { units. Determine } \vec{\mathbf{C}} \text { if }}\end{array} $$ $$ \begin{array}{l}{\text { (a) } \mathbf{C}=\mathbf{A}+\mathbf{B},(b) \mathbf{C}=\mathbf{A}-\mathbf{B},(c) \mathbf{C}=\mathbf{B}-\mathbf{A} \text { . Give the }} \\ {\text { magnitude and direction for cach. }}\end{array} $$

Zachary Warner

Numerade Educator

Problem 7

$$ 835 \mathrm{km} / \mathrm{h} \text { in a direction } 41.5^{\circ} $$ of north (Fig. $37 ) .$ (a) Find
the components of the velocity vector in the northerly and westerly directions.
(b) How far north and how far west has the plane traveled after 2.50 $\mathrm{h}$ ?

Averell Hause

Carnegie Mellon University

Problem 8

$$ \vec{\mathbf{v}}_{1}=-6.0 \hat{\mathrm{i}}+8.0 \hat{\mathrm{j}} \text { and } \vec{\mathbf{v}}_{2}=4.5 \hat{\mathrm{i}}-5.0 \mathrm{j} . $$ mine the magnitude and direction of $$
(a) \vec{\mathbf{v}}_{1},(b) \vec{\mathbf{v}}_{2} $$ $$
(c) \vec{\mathbf{v}}_{1}+\vec{\mathbf{v}}_{2} \text { and }(d) \vec{\mathbf{v}}_{2}-\vec{\mathbf{v}}_{1-}
$$

Zachary Warner

Numerade Educator

Problem 9

(II) (a) Determine the magnitude and direction of the sum of the three vectors $$
\vec{\mathbf{v}}_{1}=4.0 \hat{\mathbf{i}}-8.0 \hat{\mathbf{j}} \cdot \vec{\mathbf{v}}_{2}=\hat{\mathbf{i}}+\hat{\mathbf{j}} $$ $$ \vec{\mathbf{v}}_{3}=-2.0 \hat{\mathbf{i}}+4.0 \hat{\mathbf{j}} .(b) \text { Determine } \vec{\mathbf{v}}_{1}-\vec{\mathbf{v}}_{2}+\vec{\mathbf{v}}_{3} $$

Averell Hause

Carnegie Mellon University

Problem 10

(II) Three vectors are shown in Fig. $38 .$ Their magnitudes are given in arbitrary units. Determine the sum of the three vectors.Give the resultant in terms of $(a)$ components, (b) magnitude and angle with $x$ axis.

Zachary Warner

Numerade Educator

Problem 11

(II) (a) Given the vectors $\vec{\mathbf{A}}$ and $\vec{\mathbf{B}}$ shown in Fig. 38 , deter-
mine $\vec{\mathbf{B}}-\overline{\mathbf{A}}$ (b) Determine $\vec{\mathbf{A}}-\vec{\mathbf{B}}$ without using your answer in $(a) .$ Then compare your results and sce if they are opposite.

Averell Hause

Carnegie Mellon University

Problem 12

(II) Determine the vector $\vec{\mathbf{A}}-\vec{\mathbf{C}},$ given the vectors $\vec{\mathbf{A}}$ and $\vec{\mathbf{C}}$ in Fig. $38 .$

Zachary Warner

Numerade Educator

Problem 13

(II) For the vectors shown in Fig. $38,$ determine $(a) \vec{\mathbf{B}}-2 \vec{\mathbf{A}}$ ,
(b) $2 \vec{\mathbf{A}}-3 \vec{\mathbf{B}}+2 \vec{\mathbf{C}}$

Averell Hause

Carnegie Mellon University

Problem 14

(II) For the vectors given in Fig. 38, determine $$
(a) \vec{\mathbf{A}}-\vec{\mathbf{B}}+\vec{\mathbf{C}},(b) \overline{\mathbf{A}}+\vec{\mathbf{B}}-\vec{\mathbf{C}}, \text { and }(c) \overline{\mathbf{C}}-\vec{\mathbf{A}}-\vec{\mathbf{B}}
$$

Zachary Warner

Numerade Educator

Problem 15

(II) The summit of a mountain, 2450 $\mathrm{m}$ above base camp, is measured on a map to be 4580 $\mathrm{m}$ horizontally from the camp in a direction $32.4^{\circ}$ west of north. What are the
components of the displacement vector from camp to summit? What is its magnitude? Choose the $x$ axis cast, $y$ axis north, and $z$ axis up.

Averell Hause

Carnegie Mellon University

Problem 16

(III) You are given a vector in the $x y$ plane that has a magnitude of 90.0 units and a $y$ component of $-55.0$ units.
(a) What are the two possibilities for its $x$ component?
(b) Assuming the $x$ component is known to be positive, specify the vector which, if you add it to the original one, would give a resultant vector that is 80.0 units long and points cntirely in the $-x$ dircction.

Zachary Warner

Numerade Educator

Problem 17

(I) The position of a particular particle as a function of time is given by $$
\vec{\mathbf{r}}=\left(9.60 t \hat{\mathbf{i}}+8.85 \hat{\mathbf{j}}-1.00 t^{2} \hat{\mathbf{k}}\right) \mathrm{m} $$ Determine the particles velocity and acceleration as a function of time.

Averell Hause

Carnegie Mellon University

Problem 18

(1) What was the average velocity of the particle in Problem 17 between $t=1.00 \mathrm{s}$ and $t=3.00 \mathrm{s} ?$ What is the magnitude of the instantancous velocity at $t=2.00 \mathrm{s} ?$

Zachary Warner

Numerade Educator

Problem 19

(1) What was the average velocity of the particle in Problem 17 between $t=1.00 \mathrm{s}$ and $t=3.00 \mathrm{s} ?$ What is the magnitude of the instantancous velocity at $t=2.00 \mathrm{s}$ ?

Averell Hause

Carnegie Mellon University

Problem 20

(II) A car is moving with speed 18.0 $\mathrm{m} / \mathrm{s}$ due south at one moment and 27.5 $\mathrm{m} / \mathrm{s}$ due east 8.00 $\mathrm{s}$ later. Over this time interval, determine the magnitude and direction of $(a)$ its $$
\begin{array}{l}{\text { average velocity, }(b) \text { its average acceleration. }(c) \text { What is its }} \\ {\text { average speed. [Hint: Can you determine all these from the }} \\ {\text { information given? }}\end{array}
$$

Zachary Warner

Numerade Educator

Problem 21

(II) At $t=0,$ a particle starts from rest at $x=0, y=0$ , and moves in the $x y$ planc with an acceleration
$$ \begin{array}{l}{\vec{\mathbf{a}}=(4.0 \hat{\mathbf{i}}+3.0 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^{2} . \text { Determine }(a) \text { the } x \text { and } y \text { compo- }} \\ {\text { nents of velocity, }(b) \text { the spced of the particle, and }(c) \text { the }}\end{array} $$
position of the particle, all as a function of time. (d) Eval-
uate all the above at $t=2.0 \mathrm{s}$ .

Averell Hause

Carnegie Mellon University

Problem 22

(II) $(a)$ A skier is accelerating down a $30.0^{\circ}$ hill at 1.80 $\mathrm{m} / \mathrm{s}^{2}$
(Fig. $39 ) .$ What is the vertical component of her acceleration? (b) How long will it take her to reach the bottom of the hill, assuming she starts from rest and accelerates uniformly, if the elevation change is 325 $\mathrm{m} ?$

Zachary Warner

Numerade Educator

Problem 23

(1I) An ant walks on a piece of graph paper straight along the
$x$ axis a distance of 10.0 $\mathrm{cm}$ in 2.00 $\mathrm{s}$ . It then turns left $30.0^{\circ}$
and walks in a straight line another 10.0 $\mathrm{cm}$ in 1.80 s Finally,
it turns another $70.0^{\circ}$ to the left and walks another 10.0 $\mathrm{cm}$
in 1.55 $\mathrm{s}$ . Determine $(a)$ the $x$ and $y$ components of the ant's average velocity, and $(b)$ its magnitude and direction.

Averell Hause

Carnegie Mellon University

Problem 24

(II) A particle starts from the origin at $t=0$ with an initial
velocity of 5.0 $\mathrm{m} / \mathrm{s}$ along the positive $x$ axis If the acoclera-
tion is $(-3.0 \hat{\mathrm{i}}+4.5 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}^{2}$ , determine the velocity and position of the particle at the moment it reaches its maximum $x$ coordinate.

Zachary Warner

Numerade Educator

Problem 25

(II) Suppose the position of an object is given by $\vec{\mathbf{r}}=\left(3.0 t^{2} \hat{\mathbf{i}}-6.0 t^{3} \hat{\mathbf{j}}\right) \mathrm{m}$ (a) Determine its velocity $\vec{\mathbf{v}}$ and acceleration $\vec{a},$ as a function of time. (b) Determine $\vec{\mathbf{r}}$ and $\vec{\mathbf{v}}$ at time $t=2.5 \mathrm{s} .$

Averell Hause

Carnegie Mellon University

Problem 26

(II) An object, which is at the origin at time $t=0,$ has $$
\overline{\mathbf{v}}_{0}=(-14.0 \hat{\mathbf{i}}-7.0 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s} $$
$$ \quad \overline{\mathbf{a}}=(6.0 \hat{\mathbf{i}}+3.0 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^{2} . \text { Find the position } \overline{\mathbf{r}} $$ where the object comes to rest (momentarily).

Zachary Warner

Numerade Educator

Problem 27

(II) A particle's position as a function of time $t$ is given $$
\vec{\mathbf{r}}=\left(5.0 t+6.0 t^{2}\right) \mathrm{m} \mathbf{i}+\left(7.0-3.0 t^{3}\right) \mathrm{m} \mathbf{j} . \text { At } t=5.0 \mathrm{s} $$ $$
\begin{array}{l}{\text { find the magnitude and direction of the particle's displace- }} \\ {\text { ment vector } \Delta \vec{\mathrm{r}} \text { relative to the point } \mathbf{r}_{0}=(0.0 \mathrm{i}+7.0 \mathrm{j}) \mathrm{m}}\end{array} $$

Prabhu Ramji

Numerade Educator

Problem 28

$$ \begin{array}{l}{\text { (1) A tiger leaps horizontally from a } 7.5 \text { -migh rock with a }} \\ {\text { speed of } 3.2 \mathrm{m} / \mathrm{s} \text { . How far from the base of the rock will she }} \\ {\text { land? }}\end{array} $$

Zachary Warner

Numerade Educator

Problem 29

(1) A diver running 2.3 $\mathrm{m} / \mathrm{s}$ dives out horizontally from the
edge of a vertical cliff and 3.0 s later reaches the water below. How high was the cliff and how far from its base did the diver hit the water?

Averell Hause

Carnegie Mellon University

Problem 30

(II) Estimate how much farther a person can jump on the Moon as compared to the Earth if the takeoff speed and angle are the same. The acceleration due to gravity on the Moon is one-sixth what it is on Earth.

Chasen Shaw

Numerade Educator

Problem 31

(II) A fire hose held near the ground shoots water at a speed of 6.5 $\mathrm{m} / \mathrm{s}$ . At what angle(s) should the nozzle point in order that the water land 2.5 $\mathrm{m}$ away (Fig. 40$) ?$
Why are there two different angles? Sketch the two trajectories.

Averell Hause

Carnegie Mellon University

Problem 32

(II) A ball is thrown horizontally from the roof of a building 9.0 $\mathrm{m}$ tall and lands 9.5 $\mathrm{m}$ from the base. What was the ball's initial spced?

Zachary Warner

Numerade Educator

Problem 33

(II) A football is kicked at ground level with a spced of 18.0 $\mathrm{m} / \mathrm{s}$ at an angle of $38.0^{\circ}$ to the horizontal. How much later does it hit the ground?

Averell Hause

Carnegie Mellon University

Problem 34

(II) A ball thrown horizontally at 23.7 $\mathrm{m} / \mathrm{s}$ from the roof of
a building lands 31.0 $\mathrm{m}$ from the base of the building. How
high is the building?

Zachary Warner

Numerade Educator

Problem 35

(II) A shot-putter throws the shot (mass $=7.3 \mathrm{kg}$ ) with an
initial spced of 14.4 $\mathrm{m} / \mathrm{s}$ at a $34.0^{\circ}$ angle to the borizontal.
Calculate the horizontal distance traveled by the shot if it leaves
the athlete's hand at a height of 2.10 $\mathrm{m}$ above the ground.

Averell Hause

Carnegie Mellon University

Problem 36

(II) You buy a plastic dart gun, and being a clever physics
student you decide to do a quick calculation to find
its maximum horizontal range. You shoot the gun straight
up, and it takes 4.0 s for the dart to land back at the barrel.
What is the maximum horizontal range of your gun?

Zachary Warner

Numerade Educator

Problem 37

(II) A baseball is hit with a specd of 27.0 $\mathrm{m} / \mathrm{s}$ at an angle of
$45.0^{\circ} .$ It lands on the flat roof of a $13.0-\mathrm{m}$ -tall nearby
building. If the ball was hit when it was 1.0 $\mathrm{m}$ above the
ground, what horizontal distance does it travel before it
lands on the building?

Averell Hause

Carnegie Mellon University

Problem 38

(II) A baseball is hit with a speed of 27.0 $\mathrm{m} / \mathrm{s}$ at an angle of
$45.0^{\circ} .$ It lands on the flat roof of a $13.0-\mathrm{m}$ -tall nearby building. If the ball was hit when it was 1.0 $\mathrm{m}$ above the ground, what horizontal distance does it travel before it lands on the building?

Zachary Warner

Numerade Educator

Problem 39

(II) In Example 11 of "Kinematics in Two or Three Dimen-
sions; Vectors" we chose the $x$ axis to the right and $y$ axis
up. Redo this problem by defining the $x$ axis to the left and
$y$ axis down, and show that the conclusion remains the
same-the football lands on the ground 40.5 $\mathrm{m}$ to the right
of where it departed the punter's foot.

Averell Hause

Carnegie Mellon University

Problem 40

(II) A grasshopper hops down a level road. On each hop,
the grasshoper launches itself at angle $\theta_{0}=45^{\circ}$ and
achieves a range $R=1.0 \mathrm{m}$ . What is the average hori-
zontal speed of the grasshopper as it progresses down the
road? Assume that the time spent on the ground between
hops is negligible.

Zachary Warner

Numerade Educator

Problem 41

(II) Extreme-sports enthusiasts have been known to jump off the top of El Capitan, a shecr granite cliff of height 910 $\mathrm{m}$ in Yosemite National Park. Assume a jumper runs horizontally off the top of EI Capitan with speed 5.0 $\mathrm{m} / \mathrm{s}$ and enjoys a freefall until she is 150 $\mathrm{m}$ above the valley floor, at which time she opens her parachute (Fig. 41$)$ .(a) How long is the jumper in frecfall? Ignore air resistance. $(b)$ It is important to be as far away from the cliff as possible before opening the parachute. How far from the cliff is this jumper when she opens her chute?

Averell Hause

Carnegie Mellon University

Problem 42

(II) Here is something to try at a sporting event. Show that the maximum height $h$ attained by an objcct projected into the air, such as a baseball, football, or soccer ball, is approximately given by
$h \approx 1.2 t^{2} \mathrm{m}$
where $t$ is the total time of fight for the object in seconds Assume that the object returns to the same level as that from which it was launched, as in Fig. $42 .$ For example, if you maximum height attained was $h=1.2 \times(5.0)^{2}=30 \mathrm{m}$ . The beauty of this relation is that $h$ can be determined
without knowlcdge of the launch speed $v_{0}$ or launch angle $\theta_{0}$ .

Zachary Warner

Numerade Educator

Problem 43

(1I) The pilot of an airplane traveling 170 $\mathrm{km} / \mathrm{h}$ wants to drop supplics to flood victims isolated on a patch of land 150 $\mathrm{m}$ below. The supplies should be dropped how many
seconds before the plane is directly overhead?

Averell Hause

Carnegie Mellon University

Problem 44

(11) (a) A long jumper leaves the ground at $45^{\circ}$ above the horizontal and lands 8.0 $\mathrm{m}$ away. What is her "takeoff" spced $v_{0} ?(b)$ Now she is out on a hike and comes to the left bank of a river. There is no bridge and the right bank is 10.0 $\mathrm{m}$ away horizontally and $2.5 \mathrm{m},$ vertically below. If she long jumps from the edge of the left bank at $45^{\circ}$ with the speed calculated in $(a),$ how long, or short, of the opposite bank will she land (Fig. 43$) ?$

Zachary Warner

Numerade Educator

Problem 45

(II) A high diver leaves the end of a 5.0 -m-high diving board and strikes the water 1.3 s later, 3.0 $\mathrm{m}$ beyond the cnd of the board. Considering the diver as a particle, determine
(a) her initial velocity, $\vec{\mathbf{v}}_{0} ;(b)$ the maximum height reached:
and $(c)$ the velocity $\overline{\mathbf{v}}_{\mathrm{f}}$ with which she enters the water.

Hubert Agamasu

Numerade Educator

Problem 46

(1I) A projectile is shot from the edge of a cliff 115 $\mathrm{m}$ above ground level with an initial speed of 65.0 $\mathrm{m} / \mathrm{s}$ at an angle of $35.0^{\circ}$ with the horizontal, as shown in Fig. $44 .$ (a) Determine Determine the distance $X$ of point $P$ from the base of the
vertical cliff At the instant just before the projectile hits point $P$ , vertical cliff At the instant just before the projectile hits point $P$ find $(c)$ the horizontal and the vertical components of its velocity, $(d)$ the magnitude of the velocity, and $(e)$ the angle made by the velocity vector with the horizontal. (f) Find the maximum height above the cliff top reached by the projectile.

Zachary Warner

Numerade Educator

Problem 47

(II) Suppose the kick in Example 7 of "Kincmatics in Two or Three Dimensions; Vectors" is attempted 36.0 $\mathrm{m}$ from the goalposts, whose crossbar is 3.00 $\mathrm{m}$ above the ground. If the
football is dirccted perfectly between the goalposts, will it pass over the bar and be a field goal? Show why or why not. If not, from what horizontal distance must this kick be made if it is to score?

Averell Hause

Carnegie Mellon University

Problem 48

(II) Exactly 3.0 s after a projectile is fired into the air from the
ground, it is observed to have a velocity $\vec{\mathbf{v}}=(8.6 \hat{\mathbf{i}}+4.8 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}$ where the $x$ axis is horizontal and the $y$ axis is positive
upward. Determinc $(a)$ the horizontal range of the projectile,
(b) its maximum height above the ground, and (c) its specd and angle of motion just before it strikes the ground.

Nicholas Mogoi

Numerade Educator

Problem 49

(1I) Revisit Example 9 of "Kinematics in Two or Three Dimensions; Vectors," and assume that the boy with the slingshot is below the boy in the tree (Fig. 45) and so aims upuard, directly at the boy in the tree. Show that again the boy in the trec makes the wrong move by letting go at the moment the water balloon is shot.

Averell Hause

Carnegie Mellon University

Problem 50

(1I) A stunt driver wants to make his car jump over 8 cars parked side by side below a horizontal ramp (Fig. 46$)$ .
(a) With what minimum speed must he drive off the horizontal ramp? The vertical height of the ramp is 1.5 $\mathrm{m}$ above the cars and the horizontal distance he must clear is 22 $\mathrm{m}$ . (b) If
the ramp is now tilted upward, so that "takcoff angle" is $7.0^{\circ}$
above the horizontal, what is the new minimum speed?

Zachary Warner

Numerade Educator

Problem 51

(II) A ball is thrown horizontally from the top of a cliff with initial speed $v_{0}$ (at $t=0 )$ . At any moment, its direction of motion makes an angle $\theta$ to the horizontal (Fig. 47$)$ . Derive a formula for $\theta$ as a function of time, $t,$ as the ball follows a projectile's path.

Averell Hause

Carnegie Mellon University

Problem 52

(II) At what projection angle will the range of a projectile cqual its maximum height?

Zachary Warner

Numerade Educator

Problem 53

(1I) A projectile is fired with an initial speed of 46.6 $\mathrm{m} / \mathrm{s}$ at
an angle of $42.2^{\circ}$ above the horizontal on a long flat firing range. Determine $(a)$ the maximum height reached by the projectile, (b) the total time in the air, $(c)$ the total horizontal distance covered (that is, the range), and (d) the velocity of the projectile 1.50 s after firing.

Averell Hause

Carnegie Mellon University

Problem 54

(II) An athlete executing a long jump leaves the ground at a
$27.0^{\circ}$ angle and lands 7.80 $\mathrm{m}$ away. (a) What was the takeoff
spced? (b) If this speed were increased by just 5.0$\%$ , how much longer would the jump be?

Zachary Warner

Numerade Educator

Problem 55

(III) A person stands at the base of a hill that is a straight incline making an angle $\phi$ with the horizontal (Fig. 48). For a given initial spced $v_{0},$ at what angle $\theta$ (to the horizontal)
should objects be thrown so that the distance $d$ they land up the hill is as large as possible?

Averell Hause

Carnegie Mellon University

Problem 56

(III) Derive a formula for the horizontal range $R,$ of a
projectile when it lands at a height $h$ above its initial point.
(For $h<0,$ it lands a distance $-h$ below the starting point.)
Assume it is projected at an angle $\theta_{0}$ with initial speed $v_{0}$ .

Zachary Warner

Numerade Educator

Problem 57

(1) A person going for a morning jog on the deck of a cruise
ship is running toward the bow (front) of the ship at 2.0 $\mathrm{m} / \mathrm{s}$
while the ship is moving ahead at 8.5 $\mathrm{m} / \mathrm{s}$ . What is the velocity
of the jogger relative to the water? Later, the joger is moving toward the stern (rear) of the ship. What is the jogger's velocity relative to the water now?

Averell Hause

Carnegie Mellon University

Problem 58

(I) Huck Finn walks at a speed of 0.70 $\mathrm{m} / \mathrm{s}$ across his raft (that is, he walks
perpendicular to the raft's motion relative to the shore). The raft is traveling down the
Mississippi River at a speed of 150 $\mathrm{m} / \mathrm{s}$ relative to the nver bank
(Fig. $49 ) .$ What is Huck's velocity direction) relative to the river bank?

Zachary Warner

Numerade Educator

Problem 59

(II) Determine the spced of the boat with respect to the shore in Example 14 of "Kinematics in Two or Three Dimensions; Vectors."

Averell Hause

Carnegie Mellon University

Problem 60

(11) Two planes approach cach other head-on. Each has a
speed of 780 $\mathrm{km} / \mathrm{h}$ , and they spot cach other when they are
initially 12.0 $\mathrm{km}$ apart. How much time do the pilots have to
take evasive action?

Zachary Warner

Numerade Educator

Problem 61

(II) A child, who is 45 $\mathrm{m}$ from the bank of a river, is being carricd helplessly downstream by the river's swift current of 1.0 $\mathrm{m} / \mathrm{s} .$ As the child passes a lifeguard on the river's bank, the lifeguard starts swimming in a straight line untill she reaches the child at a point downstream (Fig. 50$)$ . If the lifeguard can swim at a speed of 2.0 $\mathrm{m} / \mathrm{s}$ relative to the
water, how long does it take her to reach the child? How far downstream does the lifeguard intercept the child?

Averell Hause

Carnegie Mellon University

Problem 62

(II) A passenger on a boat moving at 1.70 $\mathrm{m} / \mathrm{s}$ on a still lake
walks up a flight of stairs at a spced of 0.60 $\mathrm{m} / \mathrm{s}$ . Fig. $51 .$ The
stairs are angled at $45^{\circ}$ pointing in the dircction of motion
as shown. Write the vector velocity of the passenger relative
to the water.

Zachary Warner

Numerade Educator

Problem 63

(II) A person in the passenger basket of a hot-air balloon throws a ball horizontally outward from the basket with spced 10.0 $\mathrm{m} / \mathrm{s}$ (Fig. 52$)$ . What initial velocity (magnitude
and direction) does the ball have relative to a person standing on the ground (a) if the hot-air balloon is rising at 5.0 $\mathrm{m} / \mathrm{s}$ relative to the ground during this throw,
(b) if the hot-air balloon is descending at 5.0 $\mathrm{m} / \mathrm{s}$ relative to the ground.

Averell Hause

Carnegie Mellon University

Problem 64

(II) An airplane is heading due south at a speed of 580 $\mathrm{km} / \mathrm{h}$ .
If a wind begins blowing from the southwest at a speed of
90.0 $\mathrm{km} / \mathrm{h}$ (average), calculate $(a)$ the velocity (magnitude
and dircction) of the plane, relative to the ground, and (b) how far from its intended position it will be after 11.0 $\mathrm{min}$ if the pilot takes no corrective action.

Zachary Warner

Numerade Educator

Problem 65

(II) In what direction should the pilot aim the plane in
Problem 64 so that it will fly due south?

Averell Hause

Carnegie Mellon University

Problem 66

(II) Two cars approach a street corner at right angles to each other (sce Fig. $35 ) .$ Car 1 travels at 35 $\mathrm{km} / \mathrm{h}$ and car 2 at 45 $\mathrm{km} / \mathrm{h}$ . What is the relative velocity of car 1 as scen by car 2$?$ What is the velocity of car 2 relative to car 1$?$

Zachary Warner

Numerade Educator

Problem 67

(1I) A swimmer is capable of swimming 0.60 $\mathrm{m} / \mathrm{s}$ in still
water. (a) If she aims her body directly across a 55 -m-wide
river whose current is 0.50 $\mathrm{m} / \mathrm{s}$ , how far downstream (from a
point opposite her starting point) will she land? (b) How
long will it take her to reach the other side?

Averell Hause

Carnegie Mellon University

Problem 68

(II) (a) At what upstream angle must the swimmer in
Problem 67 aim, if she is to arrive at a point directly across
the stream? (b) How long will it take her?

Zachary Warner

Numerade Educator

Problem 69

(II) A motorboat whose speed in still water is 3.40 $\mathrm{m} / \mathrm{s}$ must
aim upstream at an angle of $19.5^{\circ}$ (with respect to a line perpendicular to the shore) in order to travel dircctly across the stream. (a) What is the spced of the current? (b) What is the resultant speed of the boat with respect to the shore?

Averell Hause

Carnegie Mellon University

Problem 70

(II) A boat, whose speed in still water is 2.70 $\mathrm{m} / \mathrm{s}$ , must cross
$\mathrm{a} 280-\mathrm{m}$ -wide river and arrive at a point 120 $\mathrm{m}$ upstream from where it starts (Fig. 53 ). To do so, the pilot must head the boat at a $45.0^{\circ}$ upstream angle. What is the speed of the river's current?

Zachary Warner

Numerade Educator

Problem 71

(III) An airplanc, whose air speed is 580 $\mathrm{km} / \mathrm{h}$ , is supposed
to fly in a straight path $38.0^{\circ} \mathrm{N}$ of E. But a steady 72 $\mathrm{km} / \mathrm{h}$
wind is blowing from the north. In what direction should the
plane head?

Averell Hause

Carnegie Mellon University

Problem 72

$$ \vec{\mathbf{v}}_{1} \text { and } \vec{\mathbf{v}}_{2}, \text { add to a resultant } \vec{\mathbf{v}}=\vec{\mathbf{v}}_{1}+\vec{\mathbf{v}}_{2} $$
$$ \vec{\mathbf{V}}_{1} \text { and } \vec{\mathbf{V}}_{2} \text { if }(a) V=V_{1}+V_{2},(b) V^{2}=V_{1}^{2}+V_{2}^{2} $$ $$ (c) V_{1}+V_{2}=V_{1}-V_{2} $$

Zachary Warner

Numerade Educator

Problem 73

A plumber steps out of his truck, walks 66 $\mathrm{m}$ east and 35 $\mathrm{m}$ south, and then takes an elevator 12 $\mathrm{m}$ into the subbasement of a building where a bad leak is occurring. What is the displacement of the plumber relative to his truck? Give your answer in components; also give the magnitude and angles, with respect to the $x$ axis in the vertical and horizontal planc. Assume $x$ is cast, $y$ is north, and $z$ is up.

Averell Hause

Carnegie Mellon University

Problem 74

On mountainous downhill roads, cscape routes are sometimes
placed to the side of the road for trucks whose brakes might
fail. Assuming a constant upward slope of $26^{\circ}$ , calculate the
horizontal and vertical components of the acceleration of a
truck that slowed from 110 $\mathrm{km} / \mathrm{h}$ to rest in 7.0 $\mathrm{s}$ Sce Fig. $54 .$

Zachary Warner

Numerade Educator

Problem 75

A light plane is headed due south with a specd relative to
still air of 185 $\mathrm{km} / \mathrm{h}$ . After 1.00 $\mathrm{h}$ , the pilot notices that
they have covered only 135 $\mathrm{km}$ and their direction is not
south but southeast $\left(45.0^{\circ}\right) .$ What is the wind velocity?

Averell Hause

Carnegie Mellon University

Problem 76

An Olympic long jumper is capable of jumping 8.0 $\mathrm{m}$ .
Assuming his horizontal speed is 9.1 $\mathrm{m} / \mathrm{s}$ as he leaves the
ground, how long is he in the air and how high does he go?
Assume that he lands standing upright - that is, the same
way he left the ground.

Zachary Warner

Numerade Educator

Problem 77

Romeo is chucking pebbles gently up to Julict's window, and he wants the pebbles
to hit the window with only a horizontal component of velocity. He is standing at the edge of a rose garden 8.0 below her window and 9.0 $\mathrm{m}$ from the base of the wall (Fig. 55 ). How fast are the pebbles going when they hit her window?

Averell Hause

Carnegie Mellon University

Problem 78

Raindrops make an angle $\theta$ with the vertical when viewed through a moving train window (Fig. $56 ) .$ If the speed of the train is $v_{T},$ what is the speed of the raindrops in the reference frame
of the Earth in which they are assumed to fall vertically?

Zachary Warner

Numerade Educator

Problem 79

Apollo astronauts took a "nine iron" to the Moon and hit a
golf ball about 180 $\mathrm{m}$ . Assuming that the swing. launch
angle, and so on, were the same as on Earth where the same
astronaut could hit it only $32 \mathrm{m},$ estimate the acceleration
due to gravity on the surface of the Moon. (We neglect air
resistance in both cases, but on the Moon there is none.)

Averell Hause

Carnegie Mellon University

Problem 80

A hunter aims directly at a target (on the same level) 68.0 $\mathrm{m}$
away. (a) If the bullet leaves the gun at a speed of 175 $\mathrm{m} / \mathrm{s}$ ,
by how much will it miss the target? $(b)$ At what angle
should the gun be aimed so the target will be hit?

Zachary Warner

Numerade Educator

Problem 81

The diff divers of Acapulco push off horivontally from rock platforms about 35 $\mathrm{m}$ above
the water, but they must clear rocky outcrops at water level that extend out into the water 5.0 $\mathrm{m}$ from the base of the cliff directly under their launch point.
See Fig. $57 .$ What minimum pushoff spced is necessary to clear the rocks? How long are they in the air?

Averell Hause

Carnegie Mellon University

Problem 82

When Babe Ruth hit a homer over the $8.0-$ m-high right-
ficld fence 98 $\mathrm{m}$ from home plate, roughly what was the
minimum spced of the ball when it left the bat? Assume the
ball was hit 1.0 $\mathrm{m}$ above the ground and its path initially
made a $36^{\circ}$ angle with the ground.

Zachary Warner

Numerade Educator

Problem 83

The specd of a boat in still water is $v$ . The boat is to make a round trip in a river whose current travels at speed u. Derive a formula for the time needed to make a round trip of total distance $D$ if the boat makes the round trip by moving (a) upstream and back downstream, and $(b)$ directly across
the river and back. We must assume $u<v$ why?

Averell Hause

Carnegie Mellon University

Problem 84

At scrve, a tennis player aims to hit the ball horizontally.
What minimum speed is required for the ball to clear the
$0.90-$ m-high net about 15.0 $\mathrm{m}$ from the server if the ball is "launched" from a height of 2.50 $\mathrm{m}$ ? Where will the ball land if it just clears the net (and will it be "good" in the
sense that it lands within 7.0 $\mathrm{m}$ of the net)? How long will it be in the air? See Fig. $58 .$

Zachary Warner

Numerade Educator

Problem 85

Spymaster Chris, flying a constant 208 $\mathrm{km} / \mathrm{h}$ horizontally in
a low-flying helicopter, wants to drop secret documents into
her contact's open car which is traveling 156 $\mathrm{km} / \mathrm{h}$ on a
level highway 78.0 $\mathrm{m}$ below. At what angle (with the hori-
zontal) should the car be in her sights when the packet is
released (Fig. 59$) ?$

Averell Hause

Carnegie Mellon University

Problem 86

A basketball leaves a player's hands at a height of 2.10 $\mathrm{m}$
above the floor. The basket is 3.05 $\mathrm{m}$ above the floor. The
player likes to shoot the ball at a $38.0^{\circ}$ angle. If the shot is
made from a horizontal distance of 11.00 $\mathrm{m}$ and must be
accurate to $\pm 0.22 \mathrm{m}$ (horizontally), what is the range of
initial speeds allowed to make the basket?

Zachary Warner

Numerade Educator

Problem 87

A particle has a velocity of $\vec{\mathbf{v}}=(-2.0 \hat{\mathbf{i}}+3.5 t \mathbf{j}) \mathrm{m} / \mathrm{s}$ . The particle starts at $\vec{\mathbf{r}}=(1.5 \hat{\mathrm{i}}-3.1 \hat{\mathrm{j}}) \mathrm{m}$ at $t=0 .$ Give the position and acceleration as a function of time. What is
the shape of the resulting path?

Averell Hause

Carnegie Mellon University

Problem 88

A projectile is launched from ground level to the top of a cliff which is $$
\begin{array}{ll}{195 \mathrm{m}} & {\text { away }} \\ {\text { and }} & {135 \mathrm{m}}\end{array}
$$high (see Fig 60). If the projectile lands on top of the cliff 6.6 s after it is fired, find the initial velocity of the projcctile (magnitude and direction). Neglect air resistance.

Zachary Warner

Numerade Educator

Problem 89

In hot pursuit, Agent Logan of the FBI must get directly
across a $1200-$ m-wide river in minimum time. The river's
current is 0.80 $\mathrm{m} / \mathrm{s}$ , he can row a boat at $1.60 \mathrm{m} / \mathrm{s},$ and he can run 3.00 $\mathrm{m} / \mathrm{s}$ . Describe the path he should take (rowing
plus running along the shorc) for the minimum crossing
time, and determine the minimum time.

Averell Hause

Carnegie Mellon University

Problem 90

A boat can travel 2.20 $\mathrm{m} / \mathrm{s}$ in still water. (a) If the boat
points its prow directly across a stream whose current is
$1.30 \mathrm{m} / \mathrm{s},$ what is the velocity (magnitude and direction) of
the boat relative to the shorc? (b) What will be the position
of the boat, relative to its point of origin, after 3.00 $\mathrm{s}$ ?

Zachary Warner

Numerade Educator

Problem 91

A boat is traveling where there is a current of 0.20 $\mathrm{m} / \mathrm{s}$ cast.
(Fig. 61$)$ . To avoid some offshore rocks, the boat must clear
a buoy that is NNE $\left(22.5^{\circ}\right)$ and 3.0 $\mathrm{km}$ away. The boat's
speed through still water is 21 $\mathrm{m} / \mathrm{s}$ . If the boat wants to pass
the buoy 0.15 $\mathrm{km}$ on its right, at what angle should the boat
head?

Averell Hause

Carnegie Mellon University

Problem 92

A child runs down a $12^{\circ}$ hill and then suddenly jumps upward
at a $15^{\circ}$ angle above horizontal and lands 1.4 $\mathrm{m}$ dow the hill
as measured along the hill. What was the child's initial specd?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 93

A basketball is shot from an initial height of 2.4 $\mathrm{m}$ (Fig. 62 ) with an initial speed $v_{5}=12 \mathrm{m} / \mathrm{s}$ directed at an angle $\theta_{0}=35^{\circ}$ above the horizontal. $(a)$ How far from the basket was the player if he made a basket? (b) At what angle to the horizontal did the ball enter the basket?

Averell Hause

Carnegie Mellon University

Problem 94

You are driving south on a highway at 25 $\mathrm{m} / \mathrm{s}$ (approxi-
mately 55 $\mathrm{mi} / \mathrm{h}$ ) in a snowstorm. When you last stopped, you
noticed that the snow was coming down vertically, but it is passing the windows of the moving car at an angle of $37^{\circ}$ to the horizontal. Estimate the specd of the snowflakes relative to the car and relative to the ground.

Zachary Warner

Numerade Educator

Problem 95

A rock is kicked horizontally at 15 $\mathrm{m} / \mathrm{s}$ from a hill with a
$45^{\circ}$ slope $(\mathrm{Fig}$ . 63$) .$ How long does it take for the rock to hit
the ground?

Averell Hause

Carnegie Mellon University

Problem 96

A batter hits a fly ball which leaves the bat 0.90 $\mathrm{m}$ above the
ground at an angle of $61^{\circ}$ with an initial speed of 28 $\mathrm{m} / \mathrm{s}$ head-
ing toward centerficld. Ignore air resistance. (a) How far from
home plate would the ball land if not caught? (b) The ball iscaught by the centerficlder who, starting at a distance of 105 $\mathrm{m}$ from home plate, runs straight toward home plate at a constant
speed and makes the catch at ground level. Find his spced.

Zachary Warner

Numerade Educator

Problem 97

A ball is shot from the top of a building with an initial
velocity of 18 $\mathrm{m} / \mathrm{s}$ at an angle $\theta=42^{\circ}$ above the horizontal.
(a) What are the horizontal and vertical components of the initial velocity? (b) If a nearby building is the same height and 55 $\mathrm{m}$ away, how far below the top of the building will the ball strike the nearby building?

Averell Hause

Carnegie Mellon University

Problem 98

At $t=0$ a batter hits a bascball with an initial spced of 28 $\mathrm{m} / \mathrm{s}$
at a $55^{\circ}$ angle to the horizontal. An outficlder is 85 $\mathrm{m}$ from
the batter at $t=0$ and, as seen from home plate, the line of
sight to the outficlder makes a horizontal angle of $22^{\circ}$ with the plane in which the ball moves (sce Fig. $64 ) .$ What speed and direction must the ficlder take to catch the ball at the same
height from which it was struck? Give the angle with respect to the outficlder's line of sight to home plate.

Zachary Warner

Numerade Educator

Problem 99

(II) Students shoot a plastic ball horizontally from a projectile launcher. They measure the distance $x$ the ball travels horizontally, the distance $y$ the ball falls vertically, and the total time $t$ the ball is in the air for six different heights of the projectile launcher. Here is their data.
(a) Determine the best-fit straight line that represents $x$ as a function of $t .$ What is the initial speed of the ball obtained from the best-fit straight line?
(b) Determine the best-fit quadratic cquation that represents $y$ as a function of $t$ t. What is the acceleration of the ball in the vertical direction?

Averell Hause

Carnegie Mellon University

Problem 100

(III) A shot-putter throws from a height $h=2.1 \mathrm{m}$ above
the ground as shown in Fig. $65,$ with an initial speed of
$u_{0}=13.5 \mathrm{m} / \mathrm{s} .$ (a) Derive a relation that describes how the
distance traveled $d$ depends on the release angle $\theta_{0}$ .
(b) Using the given values for $v_{0}$ and $h$ , use a graphing
calculator or computer to plot $d$ vs $\theta_{0}$ . According to your
plot, what value for $\theta_{0}$ maximizes $d$ ?

Zachary Warner

Numerade Educator