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A sailboat sails 2km due east, 5km $ 37^\circ $ south of east, and finally an unknown displacement. If the final displacement of the boat from the starting point is 6km due east, the third displacement is..........

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A sail boat sails 2 km due East, 5 km 37° South of East and finally has an unknown displacement. If the final displacement of the boat from starting point is 6 km due East, then the magnitude of third displacement

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A sail boat sails 2 km due East, 5 km  37 0  South of East and finally an unknown displacement. If the final displacement of the boat from the starting point is 6 km due East, determine the third displacement.

3 km, North

4 km, South

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Solving time: 3 mins

A sail boat sails 2   km due east, 5   km 3 7 ∘ south of east and finally an unknown displacement. If the total displacement of the boat from the starting point is 6   km due east, determine the third displacement. Given: sin 3 7 ∘ = 3/5

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Physics for JEE Main and Advanced Mechanics 1 (McGraw Hill)

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College Physics With an Integrated Approach to Forces and Kinematics

Alan giambattista, betty mccarthy richardson , robert c. richardson, acceleration and newton's second law of motion - all with video answers.

Chapter Questions

Two vectors, each of magnitude $4.0 \mathrm{~cm}$, are inclined at a small angle $\alpha$ below the horizontal as shown. Let $\overrightarrow{\mathbf{D}}=\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$. Sketch the direction of $\overrightarrow{\mathbf{D}}$ and estimate its magnitude. ( Wutorial: vectors) \begin{tabular}{|c|c|c|c|c|c|} \hline $4.0 \mathrm{~cm}$ & $\alpha_{1}$ & $\overrightarrow{\mathbf{A}}$ & & & $\overrightarrow{\mathbf{B}}$ & & $4.0 \mathrm{~cm}$ \\ \hline \end{tabular}

Vector $\overrightarrow{\mathbf{A}}$ is directed along the positive $y$ -axis and has magnitude $\sqrt{3.0}$ units. Vector $\overrightarrow{\mathbf{B}}$ is directed along the negative $x$ -axis and has magnitude $1.0$ unit. (a) What are the magnitude and direction of $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} ?(\mathrm{~b})$ What are the $x$ - and $y$ -components of $\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}$ ?

Vector $\overrightarrow{\mathbf{B}}$ has magnitude $7.1$ and direction $14^{\circ}$ below the $+x$ -axis. Vector $\overrightarrow{\mathbf{C}}$ has $x$ -component $C_{x}=-1.8$ and $y$ component $C_{y}=-6.7 .$ Compute: (a) the $x$ - and $y$ components of $\overrightarrow{\mathbf{B}} ;$ (b) the magnitude and direction of $\overrightarrow{\mathbf{C}}$; (c) the magnitude and direction of $\overrightarrow{\mathbf{C}}-\overrightarrow{\mathbf{B}} ;(\mathrm{d})$ the $x$ - and $y$ -components of $\overrightarrow{\mathbf{C}}-\overrightarrow{\mathbf{B}}$.

Margaret walks to the store using the following path: $0.500$ miles west, $0.200$ miles north, $0.300$ miles east. What is her total displacement? That is, what is the length and direction of the vector that points from her house directly to the store?

Jerry bicycles from his dorm to the local fitness center: $3.00$ miles east and $2.00$ miles north. Cindy's apartment is located $1.50$ miles west of Jerry's dorm. If Cindy is able to meet Jerry at the fitness center by bicycling in a straight line, what is the length and direction she must travel?

A runner is practicing on a circular track that is $300 \mathrm{~m}$ in circumference. From the point farthest to the west on the track, he starts off running due north and follows the track as it curves around toward the east. (a) If he runs halfway around the track and stops at the farthest eastern point of the track, what is the distance he traveled? (b) What is his displacement?

Michaela is planning a trip in Ireland from Killarney to Cork to visit Blarney Castle (see Example 3.2). She also wants to visit Mallow, which is located $39 \mathrm{~km}$ due east of Killarney and $22 \mathrm{~km}$ due north of Cork. Draw the displacement vectors for the trip when she travels from Killarney to Mallow to Cork. (a) What is the magnitude of her displacement once she reaches Cork? (b) How much additional distance does Michaela travel in going to Cork by way of Mallow instead of going directly from Killarney to Cork?

A scout troop is practicing its orienteering skills with map and compass. First they walk due east for $1.2 \mathrm{~km}$. Next, they walk $45^{\circ}$ west of north for $2.7 \mathrm{~km}$. In what direction must they walk to go directly back to their starting point? How far will they have to walk? Use graph paper, ruler, and protractor to find a geometrical solution.

Repeat Problem 8 using the component (algebraic) method.

A sailboat sails from Marblehead Harbor directly east for 45 nautical miles, then $60^{\circ}$ south of east for $20.0$ nautical miles, returns to an easterly heading for $30.0$ nautical miles, and sails $30^{\circ}$ east of north for $10.0$ nautical miles, then west for 62 nautical miles. At that time the boat becomes becalmed and the auxiliary engine fails to start. The crew decides to notify the Coast Guard of their position. Using graph paper, ruler, and protractor, find the sailboat's displacement from the harbor. Then add the displacement vectors using the component method to obtain a more accurate description of their location.

You will be hiking to a lake with some of your friends by following the trails indicated on a map at the trailhead. The map says that you will travel $1.6 \mathrm{mi}$ directly north, then $2.2$ mi in a direction $35^{\circ}$ east of north, then finally $1.1$ mi in a direction $15^{\circ}$ north of east. At the end of this hike, how far will you be from where you started, and what direction will you be from your starting point?

The pilot of a small plane finds that the airport where he intended to land is fogged in. He flies $55 \mathrm{mi}$ west to another airport to find that conditions there are too icy for him to land. He flies $25 \mathrm{mi}$ at $15^{\circ}$ east of south and is finally able to land at the third airport. (a) How far and in what direction must he fly the next day to go directly to his original destination? (b) How many extra miles beyond his original flight plan has he flown?

We have assumed that the displacement for a trip is equal to the sum of the displacements for each leg of the trip. Prove that this is true. [Hint: Imagine a trip that consists of $n$ segments. The trip starts at position $\overrightarrow{\mathbf{r}}_{1}$, proceeds to $\overrightarrow{\mathbf{r}}_{2}$, then to $\overrightarrow{\mathbf{r}}_{3}, \ldots$, then to $\overrightarrow{\mathbf{r}}_{n-1}$, then finally

A cyclist travels $10.0 \mathrm{~km}$ east in a time of $11 \min 40 \mathrm{~s}$. What is his average velocity in meters per second?

If a shot put traveled a linear distance of $23.12 \mathrm{~m}$ due north in a time of $1.65 \mathrm{~s}$, what was the average velocity of the shot?

In a game against the White Sox, baseball pitcher Nolan Ryan threw a pitch measured at $45.1 \mathrm{~m} / \mathrm{s}$. If it was $18.4 \mathrm{~m}$ from Nolan's position on the pitcher's mound to home plate, how long did it take the ball to get to the batter waiting at home plate? Treat the ball's velocity as constant and ignore any gravitational effects.

A ball thrown by a pitcher on a women's softball team is timed at $65.0 \mathrm{mph}$. The distance from the pitching rubber to home plate is $43.0 \mathrm{ft}$. In major league baseball the corresponding distance is $60.5 \mathrm{ft}$. If the batter in the softball game and the batter in the baseball game are to have equal times to react to the pitch, with what speed must the baseball be thrown? [Hint: There is no need to convert units; set up a ratio.]

See Problem 7. During Michaela's travel from Killarney to Cork via Mallow, her actual travel time in the car is 48 min. (a) What is her average speed in $\mathrm{m} / \mathrm{s}$ ? (b) What is the magnitude of her average velocity in $\mathrm{m} / \mathrm{s}$ ?

For the train in Fig. $3.2$ and Example $3.3$, find the average velocity between $3: 14$ P.M. when the train is at $3 \mathrm{~km}$ east of the origin and $3: 28$ P.M. when it is $10 \mathrm{~km}$ east of the origin.

Jason drives due west with a speed of $35.0 \mathrm{mi} / \mathrm{h}$ for $30.0 \mathrm{~min}$, then continues in the same direction with a speed of $60.0 \mathrm{mi} / \mathrm{h}$ for $2.00 \mathrm{~h}$, then drives farther west at $25.0 \mathrm{mi} / \mathrm{h}$ for $10.0 \mathrm{~min}$. What is Jason's average velocity?

Two cars, a Porsche Boxster convertible and a Toyota Scion $\mathrm{xB}$, are traveling in the same direction, although the Boxster is $186 \mathrm{~m}$ behind the Scion. The speed of the Boxster is $24.4 \mathrm{~m} / \mathrm{s}$ and the speed of the Scion is $18.6 \mathrm{~m} / \mathrm{s}$. How much time does it take for the Boxster to catch the Scion? [Hint: What must be true about the displacement of the two cars when they meet?] ( Worial: catchup)

To get to a concert in time, a harpsichordist has to drive $122 \mathrm{mi}$ in $2.00 \mathrm{~h}$. (a) If he drove at an average speed of $55.0 \mathrm{mi} / \mathrm{h}$ in a due west direction for the first $1.20 \mathrm{~h}$, what must be his average speed if he is heading $30.0^{\circ}$ south of west for the remaining $48.0$ min? (b) What is his average velocity for the entire trip?

Peggy drives from Cornwall to Atkins Glen in $45 \mathrm{~min}$. Cornwall is $73.6 \mathrm{~km}$ from Illium in a direction $25^{\circ}$ west of south. Atkins Glen is $27.2 \mathrm{~km}$ from Illium in a direction $15^{\circ}$ south of west. Using Illium as your origin, (a) draw the initial and final position vectors, (b) find the displacement during the trip, and (c) find Peggy's average velocity for the trip.

Speedometer readings are obtained and graphed as a car comes to a stop along a straight-line path. How far does the car move between $t=0$ and $t=16 \mathrm{~s} ?(\mathrm{~W}$ tutorial: start/stop traffic)

The graph shows speedometer readings, in meters per second (on the vertical axis), obtained as a skateboard travels along a straight-line path. How far does the board move between $t=3.00 \mathrm{~s}$ and $t=8.00 \mathrm{~s}$ ?

The graph shows values of $x(t)$ in meters, on the vertical axis, for a skater traveling in a straight line. (a) What is $v_{\mathrm{av}, x}$ for the interval from $t=0$ to $t=4.0 \mathrm{~s} ?$ (b) from $t=0$ to $t=5.0 \mathrm{~s}$ ?

The graph shows values of $x(t)$ in meters for a skater traveling in a straight line. What is $v_{x}$ at $t=2.0 \mathrm{~s}$ ?

The graph shows values of $x(t)$ in meters for an object traveling in a straight line. Plot $v_{x}$ as a function of time for this object from $t=0$ to $t=8 \mathrm{~s}$.

A graph is plotted of the vertical velocity component of an elevator versus time. (a) How high is the elevator above the starting point $(t=0)$ after $20 \mathrm{~s}$ have elapsed? (b) When is the elevator at its highest location above the starting point?

To pass a physical fitness test, Massimo must run $1000 \mathrm{~m}$ at an average rate of $4.0 \mathrm{~m} / \mathrm{s}$. He runs the first $900 \mathrm{~m}$ in $250 \mathrm{~s}$. Is it possible for Massimo to pass the test? If so. how fast must he run the last $100 \mathrm{~m}$ to pass the test?

A speedboat heads west at $108 \mathrm{~km} / \mathrm{h}$ for $20.0$ min. It then travels at $60.0^{\circ}$ south of west at $90.0 \mathrm{~km} / \mathrm{h}$ for $10.0$ min. (a) What is the average speed for the trip? (b) What is the average velocity for the trip?

A car travels east at $96 \mathrm{~km} / \mathrm{h}$ for $1.0 \mathrm{~h}$. It then travels $30.0^{\circ}$ east of north at $128 \mathrm{~km} / \mathrm{h}$ for $1.0 \mathrm{~h}$. (a) What is the average speed for the trip? (b) What is the average velocity for the trip?

A bicycle travels $3.2 \mathrm{~km}$ due east in $0.10 \mathrm{~h}$, then $4.8 \mathrm{~km}$ at $15.0^{\circ}$ east of north in $0.15 \mathrm{~h}$, and finally another $3.2 \mathrm{~km}$ due east in $0.10 \mathrm{~h}$ to reach its destination. The time lost in turning is negligible. What is the average velocity for the entire trip?

A motor scooter travels east at a speed of $12 \mathrm{~m} / \mathrm{s}$. The driver then reverses direction and heads west at $15 \mathrm{~m} / \mathrm{s}$. What is the change in velocity of the scooter? Give magnitude and direction.

A relay race is run along a straight-line track of length $300.0 \mathrm{~m}$ running south to north. The first runner starts at the south end of the track and passes the baton to a teammate at the north end of the track. The second runner races back to the start line and passes the baton to a third runner who races $100.0 \mathrm{~m}$ northward to the finish line. The magnitudes of the average velocities of the first, second, and third runners during their parts of the race are $7.30 \mathrm{~m} / \mathrm{s}, 7.20 \mathrm{~m} / \mathrm{s}$, and $7.80 \mathrm{~m} / \mathrm{s}$, respectively. What is the average velocity of the baton for the entire race? [Hint: You will need to find the time spent by each runner in completing her portion of the race.]

If the peak force on a runner's foot during a race is found to be vertical and three times his weight, what is the peak force on the foot of a runner whose mass is $85 \mathrm{~kg}$ ?

(a) In Fig. 3.23, what is the instantaneous acceleration of the sports car of Example $3.7$ at the time of $14 \mathrm{~s}$ from the start? (b) What is the displacement of the car from $t=12.0 \mathrm{~s}$ to $t=16.0 \mathrm{~s} ?(\mathrm{c})$ What is the average velocity of the car in the $4.0$ s time interval from $12.0 \mathrm{~s}$ to $16.0 \mathrm{~s}$ ?

If a pronghorn antelope accelerates from rest in a straight line with a constant acceleration of $1.7 \mathrm{~m} / \mathrm{s}^{2}$, how long does it take for the antelope to reach a speed of $22 \mathrm{~m} / \mathrm{s}$ ?

If a car traveling at $28 \mathrm{~m} / \mathrm{s}$ is brought to a full stop in $4.0 \mathrm{~s}$ after the brakes are applied, find the average acceleration during braking.

The graph shows $v_{x}$ versus $t$ for a body moving along a straight line. (a) What is $a_{x}$ at $t=11 \mathrm{~s}$ ? (b) What is $a_{x}$ at $$ \begin{array}{l} t=3 \mathrm{~s} ?(\mathrm{c}) \text { How far does the body travel from } t=12 \mathrm{~s} \text { to }\\ t=14 \mathrm{~s} ?(\mathrm{~W} \text { tutorial: } x, v, a) \end{array} $$

The graph with Problem 24 shows speedometer readings as a car comes to a stop. What is the magnitude of the acceleration at $t=7.0 \mathrm{~s}$ ?

The figure shows a plot of $v_{x}(t)$ for a car traveling in a straight line. (a) What is $a_{\text {av } x}$ between $t=6 \mathrm{~s}$ and $t=11 \mathrm{~s}$ ? (b) What is $v_{\mathrm{av}, x}$ for the same time interval? (c) What is $v_{\mathrm{av}, x}$ for the interval $t=0$ to $t=20 \mathrm{~s}$ ? (d) What is the increase in the car's speed between $10 \mathrm{~s}$ and $15 \mathrm{~s}$ ? (e) How far does the car travel from time $t=10 \mathrm{~s}$ to time $t=15 \mathrm{~s}$ ?

An $1100-\mathrm{kg}$ airplane starts from rest and accelerates forward for $8.0 \mathrm{~s}$ until it reaches its takeoff speed of $35 \mathrm{~m} / \mathrm{s}$. What is the average forward force on the airplane during this time? ( Wutorial: sprinter)

At $t=0$, an automobile traveling north begins to make a turn. It follows one-quarter of the arc of a circle of radius $10.0 \mathrm{~m}$ until, at $t=1.60 \mathrm{~s}$, it is traveling east. The car does not alter its speed during the turn. Find (a) the car's speed, (b) the change in its velocity during the turn, and (c) its average acceleration during the turn.

A car travels three-quarters of the way around a circle of radius $20.0 \mathrm{~m}$ in a time of $3.0 \mathrm{~s}$ at a constant speed. The initial velocity is west and the final velocity is south. (a) Find its average velocity for this trip. (b) What is the car's average acceleration during these $3.0 \mathrm{~s}$ ? (c) Explain how a car moving at constant speed has a nonzero average acceleration.

John drives $16 \mathrm{~km}$ directly west from Orion to Chester at a speed of $90 \mathrm{~km} / \mathrm{h}$, then directly south for $8.0 \mathrm{~km}$ to Seiling at a speed of $80 \mathrm{~km} / \mathrm{h}$, then finally $34 \mathrm{~km}$ southeast to Oakwood at a speed of $100 \mathrm{~km} / \mathrm{h}$. (a) What was the change in velocity during this trip? (b) What was the average acceleration during this trip?

At the beginning of a 3.0-h plane trip, you are traveling due north at $192 \mathrm{~km} / \mathrm{h}$. At the end, you are traveling $240 \mathrm{~km} / \mathrm{h}$ in the northwest direction $\left(45^{\circ}\right.$ west of north). (a) Draw your initial and final velocity vectors. (b) Find the change in your velocity. (c) What is your average acceleration during the trip?

A particle experiences a constant acceleration that is north at $100 \mathrm{~m} / \mathrm{s}^{2}$. At $t=0$, its velocity vector is $60 \mathrm{~m} / \mathrm{s}$ east. At what time will the magnitude of the velocity be $100 \mathrm{~m} / \mathrm{s} ?$

A particle experiences a constant acceleration that is south at $2.50 \mathrm{~m} / \mathrm{s}^{2}$. At $t=0$, its velocity is $40.0 \mathrm{~m} / \mathrm{s}$ east. What is its velocity at $t=8.00 \mathrm{~s}$ ?

A $1200-\mathrm{kg}$ airplane starts from rest and, with a constant acceleration of magnitude $5.0 \mathrm{~m} / \mathrm{s}^{2}$, reaches its takeoff speed in $9.00 \mathrm{~s}$. (a) What is the magnitude of the net force on the airplane during this time? (b) What is its takeoff speed?

A rubber ball is attached to a paddle by a rubber band. The ball is initially moving away from the paddle with a speed of $4.0 \mathrm{~m} / \mathrm{s}$. After $0.25 \mathrm{~s}$, the ball is moving toward the paddle with a speed of $3.0 \mathrm{~m} / \mathrm{s}$. What is the average acceleration of the ball during that $0.25 \mathrm{~s}$ ? Give magnitude and direction.

In Example $3.10$, find $T_{3}$, the tension in the coupling between cars 2 and $3 .$ ( We tutorial: towing a train)

In Fig. $3.26$ an astronaut is playing shuffleboard on Earth. The puck has a mass of $2.0 \mathrm{~kg}$. Between the board and puck the coefficient of static friction is $0.35$ and of kinetic friction is $0.25$. (a) If she pushes the puck with a force of $5.0 \mathrm{~N}$ in the forward direction, does the puck move? (b) As she is pushing, she trips and the force in the forward direction suddenly becomes $7.5 \mathrm{~N}$. Does the puck move? (c) If so, what is the acceleration of the puck along the board if she maintains contact between puck and stick as she regains her footing while pushing steadily with a force of $6.0 \mathrm{~N}$ on the puck? (d) She carries her game to the Moon and again pushes a moving puck with a force of $6.0 \mathrm{~N}$ forward. Will the acceleration of the puck during contact be more, the same, or less than on Earth? Explain. ( Watorial: rough table)

A $2.0-\mathrm{kg}$ toy locomotive is pulling a $1.0$ -kg caboose. The frictional force of the track on the caboose is $0.50 \mathrm{~N}$ backward along the track. If the train's acceleration forward is $3.0 \mathrm{~m} / \mathrm{s}^{2}$, what is the magnitude of the force exerted by the locomotive on the caboose?

A 2010 -kg elevator moves with an upward acceleration of $1.50 \mathrm{~m} / \mathrm{s}^{2}$. What is the tension in the cable that supports the elevator?

A $2010-\mathrm{kg}$ elevator moves with a downward acceleration of $1.50 \mathrm{~m} / \mathrm{s}^{2}$. What is the tension in the cable that supports the elevator?

While an elevator of mass $2530 \mathrm{~kg}$ moves upward, the tension in the cable is $33.6 \mathrm{kN}$. (a) What is the acceleration of the elevator? (b) If at some point in the motion the velocity of the elevator is $1.20 \mathrm{~m} / \mathrm{s}$ upward, what is the elevator's velocity $4.00$ s later?

An engine pulls a train of 20 freight cars, each having a mass of $5.0 \times 10^{4} \mathrm{~kg}$ with a constant force. The cars move from rest to a speed of $4.0 \mathrm{~m} / \mathrm{s}$ in $20.0 \mathrm{~s}$ on a straight track. Neglecting friction, what is the force with which the 10 th car pulls the 11 th one (at the middle of the train)? ( Watorial: school bus)

In Fig. 3.35, two blocks are connected by a lightweight, flexible cord that passes over a frictionless pulley. (a) If $m_{1}=3.0 \mathrm{~kg}$ and $m_{2}=5.0 \mathrm{~kg}$, what are the accelerations of each block? (b) What is the tension in the cord?

A helicopter is lifting two crates simultaneously. One crate with a mass of $200 \mathrm{~kg}$ is attached to the helicopter by a cable. The second crate with a mass of $100 \mathrm{~kg}$ is hanging below the first crate and attached to the first crate by a cable. As the helicopter accelerates upward at a rate of $1.0 \mathrm{~m} / \mathrm{s}^{2}$, what is the tension in each of the two cables?

An accelerometer-a device to measure accelerationcan be as simple as a small pendulum hanging in the cockpit. Suppose you are flying a small plane in a straight, horizontal line and your accelerometer hangs $12^{\circ}$ behind the vertical shown in the figure. What is your acceleration at that time?

A model sailboat is slowly sailing west across a pond at $0.33 \mathrm{~m} / \mathrm{s}$. A gust of wind blowing at $28^{\circ}$ south of west gives the sailboat a constant acceleration of magnitude $0.30 \mathrm{~m} / \mathrm{s}^{2}$ during a time interval of $2.0 \mathrm{~s}$. (a) If the net force on the sailboat during the $2.0$ -s interval has magnitude $0.375 \mathrm{~N}$, what is the sailboat's mass? (b) What is the new velocity of the boat after the $2.0$ -s gust of wind?

A rope is attached from a truck to a $1400-\mathrm{kg}$ car. The rope will break if the tension is greater than $2500 \mathrm{~N}$. Ignoring friction, what is the maximum possible acceleration of the truck if the rope does not break? Should the driver of the truck be concerned that the rope might break?

The vertical component of the acceleration of a sailplane is zero when the air pushes up against its wings with a force of $3.0 \mathrm{kN}$. (a) Assuming that the only forces on the sailplane are that due to gravity and that due to the air pushing against its wings, what is the gravitational force on the Earth due to the sailplane? (b) If the wing stalls and the upward force decreases to $2.0 \mathrm{kN}$, what is the acceleration of the sailplane?

A man lifts a $2.0$ -kg stone vertically with his hand at a constant upward velocity of $1.5 \mathrm{~m} / \mathrm{s}$. What is the magnitude of the total force of the stone on the man's hand?

A man lifts a $2.0$ -kg stone vertically with his hand at a constant upward acceleration of $1.5 \mathrm{~m} / \mathrm{s}^{2}$. What is the magnitude of the total force of the stone on the man's hand?

A small plane is flying directly west with an airspeed of $30.0 \mathrm{~m} / \mathrm{s}$. The plane flies into a region where the wind is blowing at $10.0 \mathrm{~m} / \mathrm{s}$ at an angle of $30^{\circ}$ to the south of west. (a) If the pilot does not change the heading of the plane, what will be the ground speed of the airplane? (b) What will be the new directional heading, relative to the ground, of the airplane? ( $\mathbf{W}$ tutorial: flight of crow)

A car is driving directly north on the freeway at a speed of $110 \mathrm{~km} / \mathrm{h}$ and a truck is leaving the freeway driving $85 \mathrm{~km} / \mathrm{h}$ in a direction that is $35^{\circ}$ west of north. What is the velocity of the truck relative to the car?

A Nile cruise ship takes $20.8 \mathrm{~h}$ to go upstream from Luxor to Aswan, a distance of $208 \mathrm{~km}$, and $19.2 \mathrm{~h}$ to make the return trip downstream. Assuming the ship's speed relative to the water is the same in both cases, calculate the speed of the current in the Nile.

Two cars are driving toward each other on a straight, flat Kansas road. The Jeep Wrangler is traveling at $82 \mathrm{~km} / \mathrm{h}$ north and the Ford Taurus is traveling at $48 \mathrm{~km} / \mathrm{h}$ south, both measured relative to the road. What is the velocity of the Jeep relative to an observer in the Ford?

Two cars are driving toward each other on a straight and level road in Alaska. The BMW is traveling at $100.0 \mathrm{~km} / \mathrm{h}$ north and the VW is traveling at $42 \mathrm{~km} / \mathrm{h}$ south, both velocities measured relative to the road. At a certain instant, the distance between the cars is $10.0 \mathrm{~km}$. Approximately how long will it take from that instant for the two cars to meet? [Hint: Consider a reference frame in which one of the cars is at rest.

An airplane has a velocity relative to the ground of $210 \mathrm{~m} / \mathrm{s}$ toward the east. The pilot measures his airspeed (the speed of the plane relative to the air) to be $160 \mathrm{~m} / \mathrm{s}$. What is the minimum wind velocity possible?

A boat that can travel at $4.0 \mathrm{~km} / \mathrm{h}$ in still water crosses a river with a current of $1.8 \mathrm{~km} / \mathrm{h}$. At what angle must the boat be pointed upstream to travel straight across the river? In other words, in what direction is the velocity of the boat relative to the water?

A boy is attempting to swim directly across a river; he is able to swim at a speed of $0.500 \mathrm{~m} / \mathrm{s}$ relative to the water. The river is $25.0 \mathrm{~m}$ wide and the boy ends up at $50.0 \mathrm{~m}$ downstream from his starting point. (a) How fast is the current flowing in the river? (b) What is the speed of the boy relative to a friend standing on the riverbank?

An aircraft has to fly between two cities, one of which is $600.0 \mathrm{~km}$ north of the other. The pilot starts from the southern city and encounters a steady $100.0 \mathrm{~km} / \mathrm{h}$ wind that blows from the northeast. The plane has a cruising speed of $300.0 \mathrm{~km} / \mathrm{h}$ in still air. (a) In what direction (relative to east) must the pilot head her plane? (b) How long does the flight take?

At an antique car rally, a Stanley Steamer automobile travels north at $40 \mathrm{~km} / \mathrm{h}$ and a Pierce Arrow automobile travels east at $50 \mathrm{~km} / \mathrm{h}$. Relative to an observer riding in the Stanley Steamer, what are the $x$ - and $y$ -components of the velocity of the Pierce Arrow car? The $x$ -axis is to the east and the $y$ -axis is to the north.

Sheena can row a boat at $3.00 \mathrm{mi} / \mathrm{h}$ in still water. She needs to cross a river that is $1.20$ mi wide with a current flowing at $1.60 \mathrm{mi} / \mathrm{h}$. Not having her calculator ready, she guesses that to go straight across, she should head $60.0^{\circ}$ upstream. (a) What is her speed with respect to the starting point on the bank? (b) How long does it take her to cross the river? (c) How far upstream or downstream from her starting point will she reach the opposite bank? (d) In order to go straight across, what angle upstream should she have headed?

A small plane is flying directly west with an airspeed of $30.0 \mathrm{~m} / \mathrm{s}$. The plane flies into a region where the wind is blowing at $10.0 \mathrm{~m} / \mathrm{s}$ at an angle of $30^{\circ}$ to the south of west. In that region, the pilot changes the directional heading to maintain her due west heading. (a) What is the change she makes in the directional heading to compensate for the wind? (b) After the heading change, what is the ground speed of the airplane?

Demonstrate with a vector diagram that a displacement is the same when measured in two different reference frames that are at rest with respect to each other.

A dolphin wants to swim directly back to its home bay, which is $0.80 \mathrm{~km}$ due west. It can swim at a speed of $4.00 \mathrm{~m} / \mathrm{s}$ relative to the water, but a uniform water current flows with speed $2.83 \mathrm{~m} / \mathrm{s}$ in the southeast direction. (a) What direction should the dolphin head? (b) How long does it take the dolphin to swim the $0.80-\mathrm{km}$ distance home?

In a plate glass factory, sheets of glass move along a conveyor belt at a speed of $15.0 \mathrm{~cm} / \mathrm{s}$. An automatic cutting tool descends at preset intervals to cut the glass to size. Since the assembly belt must keep moving at constant speed, the cutter is set to cut at an angle to compensate for the motion of the glass. If the glass is $72.0 \mathrm{~cm}$ wide and the cutter moves across the width at a speed of $24.0 \mathrm{~cm} / \mathrm{s}$, at what angle should the cutter be set?

(a) If a freestyle swimmer travels $1.50 \mathrm{~km}$ in a time of $14 \min 53 \mathrm{~s}$, how fast is his average speed? (b) If the pool is rectangular and $50 \mathrm{~m}$ in length, how does the speed you found compare with his sustained swimming speed of $1.54 \mathrm{~m} / \mathrm{s}$ during one length of the pool after he has been swimming for 10 min? What might account for the difference?

A jetliner flies east for $600.0 \mathrm{~km}$, then turns $30.0^{\circ}$ toward the south and flies another $300.0 \mathrm{~km}$. (a) How far is the plane from its starting point? (b) In what direction could the jetliner have flown directly to the same destination (in a straight-line path)? (c) If the jetliner flew at a constant speed of $400.0 \mathrm{~km} / \mathrm{h}$, how long did the trip take? (d) Moving at the same speed, how long would the direct flight have taken?

A pilot wants to fly from Dallas to Oklahoma City, a distance of $330 \mathrm{~km}$ at an angle of $10.0^{\circ}$ west of north. The pilot heads directly toward Oklahoma City with an air speed of $200 \mathrm{~km} / \mathrm{h}$. After flying for $1.0 \mathrm{~h}$, the pilot finds that he is $15 \mathrm{~km}$ off course to the west of where he expected to be after one hour assuming there was no wind. (a) What is the velocity and direction of the wind? (b) In what direction should the pilot have headed his plane to fly directly to Oklahoma City without being blown off course?

In a playground, two slides have different angles of incline $\theta_{1}$ and $\theta_{2}\left(\theta_{2}>\theta_{1}\right) .$ A child slides down the first at constant speed; on the second, his acceleration down the slide is $a$. Assume the coefficient of kinetic friction is the same for both slides. (a) Find $a$ in terms of $\theta_{1}, \theta_{2}$, and $g$. (b) Find the numerical value of $a$ for $\theta_{1}=45^{\circ}$ and $\theta_{2}=61^{\circ} .$

At 3:00 P.M., a bank robber is spotted driving north on I- 15 at milepost 126. His speed is $112.0 \mathrm{mi} / \mathrm{h}$. At 3:37 P.M., he is spotted at milepost 185 doing $105.0 \mathrm{mi} / \mathrm{h}$. During this time interval, what are the bank robber's displacement, average velocity, and average acceleration? (Assume a straight highway.)

The coefficient of static friction between a block and a horizontal floor is $0.35$, while the coefficient of kinetic friction is $0.22$. The mass of the block is $4.6 \mathrm{~kg}$ and it is initially at rest. (a) What is the minimum horizontal applied force required to make the block start to slide? (b) Once the block is sliding, if you keep pushing on it with the same minimum starting force as in part (a), does the block move with constant velocity or does it accelerate? (c) If it moves with constant velocity, what is its velocity? If it accelerates, what is its acceleration?

In a movie, a stuntman places himself on the front of a truck as the truck accelerates. The coefficient of friction between the stuntman and the truck is $0.65 .$ The stunt- man is not standing on anything but can "stick" to the front of the truck as long as the truck continues to accelerate. What minimum forward acceleration will keep the stuntman on the front of the truck?

A crate of oranges weighing $180 \mathrm{~N}$ rests on a flatbed truck $2.0 \mathrm{~m}$ from the back of the truck. The coefficients of friction between the crate and the bed are $\mu_{\mathrm{s}}=0.30$ and $\mu_{\mathrm{k}}=0.20$. The truck drives on a straight, level highway at a constant $8.0 \mathrm{~m} / \mathrm{s}$. (a) What is the force of friction acting on the crate? (b) If the truck speeds up with an acceleration of $1.0 \mathrm{~m} / \mathrm{s}^{2}$, what is the force of the friction on the crate? (c) What is the maximum acceleration the truck can have without the crate starting to slide?

A bicycle is moving along a straight line. The graph in the figure shows its position from the starting point as a function of time. (a) In which section(s) of the graph does the object have the highest speed? (b) At which time(s) does the object reverse its direction of motion? (c) How far does the object move from $t=0$ to $t=3 \mathrm{~s}$ ?

A crate of books is to be put on a truck by rolling it up an incline of angle $\theta$ using a dolly. The total mass of the crate and the dolly is $m .$ Assume that rolling the dolly up the incline is the same as sliding it up a frictionless surface. (a) What is the magnitude of the horizontal force that must be applied just to hold the crate in place on the incline? (b) What horizontal force must be applied to roll the crate up at constant speed? (c) In order to start the dolly moving, it must be accelerated from rest. What horizontal force must be applied to give the crate an acceleration up the incline of magnitude $a$ ? ( Wo tutorial: cart on ramp)

A toy cart of mass $m_{1}$ moves on frictionless wheels as it is pulled by a string under tension $T$. A block of mass $m_{2}$ rests on top of the cart. The coefficient of static friction between the cart and the block is $\mu$. Find the maximum tension $T$ that will not cause the block to slide on the cart if the cart rolls on (a) a horizontal surface: (b) up a ramp of angle $\theta$ above the horizontal. In both cases, the string is parallel to the surface on which the cart rolls.

Imagine a trip where you drive along an east-west highway at $80.0 \mathrm{~km} / \mathrm{h}$ for $45.0$ min and then you turn onto a highway that runs $38.0^{\circ}$ north of east and travel at $60.0 \mathrm{~km} / \mathrm{h}$ for $30.0$ min. (a) What is your average velocity for the trip? (b) What is your average velocity on the return trip when you head the opposite way and drive $38.0^{\circ}$ south of west at $60.0 \mathrm{~km} / \mathrm{h}$ for the first $30.0$ min and then west at $80.0 \mathrm{~km} / \mathrm{h}$ for the last $45.0 \mathrm{~min} ?$

A rocket is launched from rest. After $8.0 \mathrm{~min}$, it is $160 \mathrm{~km}$ above the Earth's surface and is moving at a speed of $7.6 \mathrm{~km} / \mathrm{s}$. Assuming the rocket moves up in a straight line, what are its (a) average velocity and (b) average acceleration?

Based on the information given in Problem 94 , is it possible that the rocket moves with constant acceleration? Explain.

To pass a physical fitness test, Marcella must run $1000 \mathrm{~m}$ at an average speed of $4.00 \mathrm{~m} / \mathrm{s}$. She runs the first $500 \mathrm{~m}$ at an average of $4.20 \mathrm{~m} / \mathrm{s}$. What should be her average speed over the last $500 \mathrm{~m}$ in order to finish with an overall average speed of $4.00 \mathrm{~m} / \mathrm{s}$ ?

A locomotive pulls a train of 10 identical cars, on a track that runs east-west, with a force of $2.0 \times 10^{6} \mathrm{~N}$ directed east. What is the force with which the last car to the west pulls on the rest of the train?

Beatrice needs to drag her suitcase again (see Example 3.8). This time she pulls with a force of $105 \mathrm{~N}$ at $38.0^{\circ}$ with the horizontal. The coefficients of static and kinetic friction between the suitcase and the floor are $0.273$ and $0.117$, respectively. (a) Draw an FBD of the suitcase. (b) What is the magnitude of the normal force acting on the suitcase due to the floor? (c) Does the suitcase slide, or does she need to increase her force? (d) If the suitcase slides, what is its acceleration? If it does not slide, to what magnitude should she increase her force?

A woman of mass $51 \mathrm{~kg}$ is standing in an elevator. (a) If the elevator floor pushes up on her feet with a force of $408 \mathrm{~N}$, what is the acceleration of the elevator? (b) If the elevator is moving at $1.5 \mathrm{~m} / \mathrm{s}$ as it passes the fourth floor on its way down, what is its speed $4.0 \mathrm{~s}$ later?

Problem 100

Two blocks lie side by side on a frictionless table. The block on the left is of mass $m$; the one on the right is of mass $2 m$. The block on the right is pushed to the left with a force of magnitude $F$, pushing the other block in turn. What force does the block on the left exert on the block to its right?

Problem 101

A person climbs from a Paris metro station to the street level by walking up a stalled escalator in $94 \mathrm{~s}$. It takes $66 \mathrm{~s}$ to ride the same distance when standing on the escalator when it is operating normally. How long would it take for him to climb from the station to the street by walking up the moving escalator?

Problem 102

A pilot starting from Athens, New York, wishes to fly to Sparta, New York, which is $320 \mathrm{~km}$ from Athens in the direction $20.0^{\circ} \mathrm{N}$ of $\mathrm{E}$. The pilot heads directly for Sparta and flies at an airspeed of $160 \mathrm{~km} / \mathrm{h}$. After flying for $2.0 \mathrm{~h}$, the pilot expects to be at Sparta, but instead he finds himself $20 \mathrm{~km}$ due west of Sparta. He has forgotten to correct for the wind. (a) What is the velocity of the plane relative to the air? (b) Find the velocity (magnitude and direction) of the plane relative to the ground. (c) Find the wind speed and direction.

Problem 103

Two blocks, masses $m_{1}$ and $m_{2}$, are connected by a massless cord. If the two blocks are pulled with a constant tension on a frictionless surface by applying a force of magnitude $T_{2}$ to a second cord connected to $m_{2}$, what is the ratio of the tensions in the two cords $T_{1} / T_{2}$ in terms of the masses?

Problem 104

The coefficient of static friction between a brick and a wooden hoard is $0.40$ and the coefficient of kinetic friction between the brick and board is $0.30$. You place the brick on the board and slowly lift one end of the board off the ground until the brick starts to slide down the board. (a) What angle does the board make with the ground when the brick starts to slide? (b) What is the acceleration of the brick as it slides down the board?

Problem 105

In the human nervous system, signals are transmitted along neurons as action potentials that travel at speeds of up to $100 \mathrm{~m} / \mathrm{s}$. (An action potential is a traveling influx of sodium ions through the membrane of a neuron.) The signal is passed from one neuron to another by the release of neurotransmitters in the synapse. Suppose someone steps on your toe. The pain signal travels along a 1.0-m-long sensory neuron to the spinal column, across a synapse to a second $1.0$ -m-long neuron, and across a second synapse to the brain. Suppose that the synapses are each $100 \mathrm{~nm}$ wide, that it takes $0.10 \mathrm{~ms}$ for the signal to cross each synapse, and that the action potentials travel at $100 \mathrm{~m} / \mathrm{s}$. (a) At what average speed does the signal cross a synapse? (b) How long does it take the signal to reach the brain? (c) What is the average speed of propagation of the signal?

Problem 106

An airplane of mass $2800 \mathrm{~kg}$ has just lifted off the runway. It is gaining altitude at a constant $2.3 \mathrm{~m} / \mathrm{s}$ while the horizontal component of its velocity is increasing at a rate of $0.86 \mathrm{~m} / \mathrm{s}^{2}$. Assume $g=9.81 \mathrm{~m} / \mathrm{s}^{2}$. (a) Find the direction of the force exerted on the airplane by the air. (b) Find the horizontal and vertical components of the plane's acceleration if the force due to the air has the same magnitude but has a direction $2.0^{\circ}$ closer to the vertical than its direction in part (a).

Problem 107

The graph shows the position $x$ of a switch engine in a rail yard as a function of time $t$. At which of the labeled times $t_{0}$ to $t_{7}$ is (a) $a_{x}<0$, (b) $a_{x}=0$, (c) $a_{x}>0$, (d) $v_{x}=0$, (e) the speed decreasing?

Problem 108

An elevator starts at rest on the ninth floor. At $t=0$, a passenger pushes a button to go to another floor. The graph for this problem shows the acceleration $a_{y}$ of the elevator as a function of time. Let the $y$ -axis point upward. (a) Has the passenger gone to a higher or lower floor? (b) Sketch a graph of the velocity $v_{y}$ of the elevator versus time. (c) Sketch a graph of the position $y$ of the elevator versus time. (d) If a $63.5-\mathrm{kg}$ person stands

Problem 109

A helicopter of mass $M$ is lowering a truck of mass $m$ onto the deck of a ship. (a) At first, the helicopter and the truck move downward together (the length of the cable doesn't change). If their downward speed is decreasing at a rate of $0.10 g$, what is the tension in the cable? (b) As the truck gets close to the deck, the helicopter stops moving downward. While it hovers, it lets out the cable so that the truck is still moving downward. If the truck's downward speed is decreasing at a rate of $0.10 g$, while the helicopter is at rest, what is the tension in the cable?

A sailboat sails 2km east, then 4\sqrt2 km northeast then an additional dis†an ce in a unknown direction. Its final position is 5 km directly east of the starting point. Find the magnitude and direction of the third leg of the journey