A Plane Takes Off and Travels at an Angle of 40 North of East

4.1 Displacement and Velocity Vectors

17.

The coordinates of a particle in a rectangular coordinate system are (i.0, –4.0, 6.0). What is the position vector of the particle?

18 .

The position of a particle changes from ${\overrightarrow{r}}_1=(2.0\text{}\widehat{i}+\mathrm{iii.0}\widehat{j})\text{cm}$ to ${\overrightarrow{r}}_2=(\mathrm{-four.0}\widehat{i}+\mathrm{iii.0}\widehat{j})\text{cm}.$ What is the particle'due south deportation?

19.

The 18th hole at Pebble Beach Golf Course is a dogleg to the left of length 496.0 yard. The fairway off the tee is taken to exist the ten direction. A golfer hits his tee shot a distance of 300.0 1000, corresponding to a displacement $\text{Δ}{\overrightarrow{r}}_{\mathrm{i}}=300.0\text{m}\widehat{i},$ and hits his 2nd shot 189.0 m with a displacement $\text{Δ}{\overrightarrow{r}}_{\mathrm{ii}}=172.0\text{m}\widehat{i}+80.3\text{m}\widehat{j}.$ What is the final displacement of the golf ball from where it started?

20 .

A bird flies straight northeast a altitude of 95.0 km for 3.0 h. With the x-axis due east and the y-centrality due north, what is the displacement in unit vector notation for the bird? What is the average velocity for the trip?

21.

A cyclist rides 5.0 km due e, then 10.0 km $20\text{°}$ west of north. From this point she rides 8.0 km due due west. What is the concluding displacement from where the cyclist started?

22 .

New York Rangers defenseman Daniel Girardi stands at the goal and passes a hockey puck twenty k and $45\text{°}$ from straight downwards the ice to left wing Chris Kreider waiting at the blue line. Kreider waits for Girardi to reach the blue line and passes the puck directly across the ice to him ten k abroad. What is the concluding deportation of the puck? Encounter the post-obit effigy.

23.

The position of a particle is $\overrightarrow{r}(t)=4.0t^2\widehat{i}-3.0\widehat{j}+2.0t^3\widehat{\mathrm{yard}}\text{m}.$ (a) What is the velocity of the particle at 0 s and at $\mathrm{ane.0}$ s? (b) What is the average velocity between 0 s and $1.0$ s?

24 .

Clay Matthews, a linebacker for the Green Bay Packers, can reach a speed of x.0 k/southward. At the beginning of a play, Matthews runs downfield at $45\text{°}$ with respect to the 50-thou line and covers 8.0 k in i south. He then runs straight downwardly the field at $\mathrm{xc}\text{°}$ with respect to the 50-yard line for 12 g, with an elapsed fourth dimension of i.2 s. (a) What is Matthews' final deportation from the kickoff of the play? (b) What is his average velocity?

25.

The F-35B Lighting 2 is a short-takeoff and vertical landing fighter jet. If it does a vertical takeoff to xx.00-m superlative above the ground and and so follows a flight path angled at $\mathrm{thirty}\text{°}$ with respect to the basis for 20.00 km, what is the terminal deportation?

4.two Acceleration Vector

26 .

The position of a particle is $\overrightarrow{r}(t)=(3.0t^{\mathrm{ii}}\widehat{i}+5.0\widehat{j}-6.0t\widehat{g})\text{m}.$ (a) Determine its velocity and acceleration every bit functions of time. (b) What are its velocity and acceleration at time t = 0?

27.

A particle'south acceleration is $(\mathrm{iv.0}\widehat{i}+3.0\widehat{j})\text{m/}{\text{s}}^2.$ At t = 0, its position and velocity are zero. (a) What are the particle'southward position and velocity every bit functions of fourth dimension? (b) Find the equation of the path of the particle. Draw the x- and y-axes and sketch the trajectory of the particle.

28 .

A boat leaves the dock at t = 0 and heads out into a lake with an acceleration of $2.0\text{m/}{\text{s}}^2\widehat{i}.$ A strong wind is pushing the boat, giving it an boosted velocity of $2.0\text{1000/s}\widehat{i}+\mathrm{i.0}\text{m/s}\widehat{j}.$ (a) What is the velocity of the boat at t = 10 s? (b) What is the position of the boat at t = 10s? Draw a sketch of the boat'southward trajectory and position at t = 10 s, showing the x- and y-axes.

29.

The position of a particle for t > 0 is given by $\overrightarrow{r}(t)=(3.0t^2\widehat{i}-\mathrm{seven.0}{t}^3\widehat{j}-5.0t^{\mathrm{-2}}\widehat{k})\text{m}.$ (a) What is the velocity as a office of fourth dimension? (b) What is the dispatch every bit a function of time? (c) What is the particle's velocity at t = two.0 s? (d) What is its speed at t = 1.0 due south and t = 3.0 southward? (e) What is the average velocity between t = 1.0 s and t = 2.0 southward?

xxx .

The acceleration of a particle is a constant. At t = 0 the velocity of the particle is $(10\widehat{i}+\mathrm{twenty}\widehat{j})\text{m/s}.$ At t = 4 southward the velocity is $\mathrm{x}\widehat{j}\text{m/s}.$ (a) What is the particle'southward acceleration? (b) How do the position and velocity vary with fourth dimension? Assume the particle is initially at the origin.

31.

A particle has a position function $\overrightarrow{r}(t)=\text{cos}(\mathrm{ane.0}t)\widehat{i}+\text{sin}(\mathrm{one.0}t)\widehat{j}+t\widehat{k},$ where the arguments of the cosine and sine functions are in radians. (a) What is the velocity vector? (b) What is the acceleration vector?

32 .

A Lockheed Martin F-35 II Lighting jet takes off from an aircraft carrier with a runway length of 90 m and a takeoff speed seventy m/s at the end of the runway. Jets are catapulted into airspace from the deck of an shipping carrier with ii sources of propulsion: the jet propulsion and the catapult. At the point of leaving the deck of the aircraft carrier, the F-35'south acceleration decreases to a constant acceleration of $5.0\text{m/}{\text{s}}^{\mathrm{ii}}$ at $\mathrm{xxx}\text{°}$ with respect to the horizontal. (a) What is the initial dispatch of the F-35 on the deck of the aircraft carrier to arrive airborne? (b) Write the position and velocity of the F-35 in unit vector notation from the point it leaves the deck of the aircraft carrier. (c) At what altitude is the fighter five.0 s afterwards it leaves the deck of the shipping carrier? (d) What is its velocity and speed at this time? (e) How far has it traveled horizontally?

iv.3 Projectile Motion

33.

A bullet is shot horizontally from shoulder height (i.5 g) with an initial speed 200 m/due south. (a) How much time elapses before the bullet hits the basis? (b) How far does the bullet travel horizontally?

34 .

A marble rolls off a tabletop 1.0 m loftier and hits the flooring at a point 3.0 m abroad from the table's edge in the horizontal management. (a) How long is the marble in the air? (b) What is the speed of the marble when it leaves the table'southward edge? (c) What is its speed when it hits the floor?

35.

A dart is thrown horizontally at a speed of ten m/s at the bull's-centre of a dartboard 2.4 thou abroad, as in the following figure. (a) How far beneath the intended target does the dart hit? (b) What does your reply tell you about how skilful sprint players throw their darts?

36 .

An airplane flying horizontally with a speed of 500 km/h at a height of 800 thou drops a crate of supplies (run into the following figure). If the parachute fails to open, how far in forepart of the release point does the crate hit the ground?

37.

Suppose the plane in the preceding problem fires a projectile horizontally in its direction of motion at a speed of 300 m/south relative to the plane. (a) How far in front of the release point does the projectile hit the ground? (b) What is its speed when it hits the basis?

38 .

A fastball pitcher can throw a baseball game at a speed of forty m/s (90 mi/h). (a) Assuming the pitcher tin can release the ball 16.7 m from home plate so the ball is moving horizontally, how long does information technology take the ball to reach dwelling house plate? (b) How far does the brawl drop between the pitcher'south hand and home plate?

39.

A projectile is launched at an angle of $30\text{°}$ and lands xx s after at the same summit as information technology was launched. (a) What is the initial speed of the projectile? (b) What is the maximum distance? (c) What is the range? (d) Calculate the displacement from the point of launch to the position on its trajectory at 15 southward.

40 .

A basketball game role player shoots toward a basket half-dozen.1 m away and three.0 m in a higher place the floor. If the ball is released 1.viii yard to a higher place the floor at an angle of $60\text{°}$ above the horizontal, what must the initial speed be if it were to go through the basket?

41.

At a detail instant, a hot air airship is 100 grand in the air and descending at a constant speed of ii.0 m/south. At this exact instant, a daughter throws a ball horizontally, relative to herself, with an initial speed of 20 m/due south. When she lands, where will she discover the ball? Ignore air resistance.

42 .

A homo on a motorcycle traveling at a uniform speed of ten g/southward throws an empty can straight upward relative to himself with an initial speed of three.0 m/s. Observe the equation of the trajectory as seen by a police officeholder on the side of the route. Assume the initial position of the can is the point where it is thrown. Ignore air resistance.

43.

An athlete tin can leap a distance of 8.0 one thousand in the wide jump. What is the maximum distance the athlete can leap on the Moon, where the gravitational acceleration is ane-sixth that of Earth?

44 .

The maximum horizontal altitude a boy can throw a ball is 50 m. Assume he tin can throw with the same initial speed at all angles. How high does he throw the brawl when he throws information technology straight upward?

45.

A rock is thrown off a cliff at an angle of $53\text{°}$ with respect to the horizontal. The cliff is 100 1000 high. The initial speed of the rock is xxx thousand/s. (a) How loftier above the edge of the cliff does the rock ascent? (b) How far has it moved horizontally when it is at maximum altitude? (c) How long afterwards the release does it hit the ground? (d) What is the range of the rock? (e) What are the horizontal and vertical positions of the stone relative to the border of the cliff at t = 2.0 s, t = 4.0 s, and t = half dozen.0 s?

46 .

Trying to escape his pursuers, a secret amanuensis skis off a slope inclined at $30\text{°}$ below the horizontal at lx km/h. To survive and land on the snow 100 m below, he must clear a gorge 60 one thousand wide. Does he go far? Ignore air resistance.

47.

A golfer on a fairway is 70 thou abroad from the dark-green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an bending of $\mathrm{forty}\text{°}$ with an initial speed of 20 m/s, how shut to the green does she come?

48 .

A projectile is shot at a hill, the base of operations of which is 300 yard away. The projectile is shot at $60\text{°}$ above the horizontal with an initial speed of 75 m/south. The colina can be approximated past a plane sloped at $20\text{°}$ to the horizontal. Relative to the coordinate organisation shown in the post-obit figure, the equation of this straight line is $y=(\text{tan}20\text{°})\mathrm{ten}-109.$ Where on the hill does the projectile country?

49.

An astronaut on Mars kicks a soccer ball at an angle of $45\text{°}$ with an initial velocity of xv one thousand/s. If the acceleration of gravity on Mars is $3.7\text{m/}{\text{s}}^{\mathrm{ii}}$ , (a) what is the range of the soccer kick on a apartment surface? (b) What would exist the range of the aforementioned kick on the Moon, where gravity is one-sixth that of Earth?

l .

Mike Powell holds the tape for the long jump of 8.95 m, established in 1991. If he left the ground at an angle of $\mathrm{fifteen}\text{°},$ what was his initial speed?

51.

MIT's robot cheetah can jump over obstacles 46 cm high and has speed of 12.0 km/h. (a) If the robot launches itself at an angle of $\mathrm{sixty}\text{°}$ at this speed, what is its maximum top? (b) What would the launch angle have to exist to attain a height of 46 cm?

52 .

Mt. Asama, Japan, is an active volcano. In 2009, an eruption threw solid volcanic rocks that landed 1 km horizontally from the crater. If the volcanic rocks were launched at an angle of $\mathrm{xl}\text{°}$ with respect to the horizontal and landed 900 thousand beneath the crater, (a) what would exist their initial velocity and (b) what is their time of flying?

53.

Drew Brees of the New Orleans Saints tin throw a football 23.0 m/s (50 mph). If he angles the throw at $10\text{°}$ from the horizontal, what distance does it get if it is to be defenseless at the aforementioned peak as information technology was thrown?

54 .

The Lunar Roving Vehicle used in NASA's late Apollo missions reached an unofficial lunar land speed of $\mathrm{five.0}\text{yard/s}$ by astronaut Eugene Cernan. If the rover was moving at this speed on a flat lunar surface and hitting a pocket-size bump that projected it off the surface at an angle of $20\text{°},$ how long would it be "airborne" on the Moon?

55.

A soccer goal is two.44 g high. A player kicks the brawl at a distance ten chiliad from the goal at an angle of $25\text{°}.$ The ball hits the crossbar at the pinnacle of the goal. What is the initial speed of the soccer ball?

56 .

Olympus Mons on Mars is the largest volcano in the solar arrangement, at a acme of 25 km and with a radius of 312 km. If yous are standing on the summit, with what initial velocity would you accept to burn a projectile from a cannon horizontally to articulate the volcano and land on the surface of Mars? Annotation that Mars has an acceleration of gravity of $\mathrm{three.seven}\text{m}\text{/}{\text{s}}^2.$

57.

In 1999, Robbie Knievel was the first to spring the Grand Canyon on a motorbike. At a narrow part of the canyon (69.0 thou wide) and traveling 35.8 m/southward off the takeoff ramp, he reached the other side. What was his launch angle?

58 .

Y'all throw a baseball at an initial speed of xv.0 grand/s at an angle of $30\text{°}$ with respect to the horizontal. What would the ball's initial speed have to be at $\mathrm{xxx}\text{°}$ on a planet that has twice the dispatch of gravity as Globe to achieve the same range? Consider launch and touch on a horizontal surface.

59.

Aaron Rodgers throws a football at twenty.0 k/due south to his broad receiver, who is running straight down the field at 9.4 k/s. If Aaron throws the football when the broad receiver is 10.0 1000 in front end of him, (a) at what angle does Aaron have to launch the brawl so the ball will be at the same height as the receiver when the receiver makes it to xx.0 one thousand in front end of Aaron? (b) Will the receiver exist able to catch the ball?

4.iv Uniform Circular Motion

60 .

A flywheel is rotating at thirty rev/s. What is the total angle, in radians, through which a point on the flywheel rotates in 40 s?

61.

A particle travels in a circle of radius x m at a constant speed of twenty chiliad/s. What is the magnitude of the dispatch?

62 .

Cam Newton of the Carolina Panthers throws a perfect football spiral at 8.0 rev/southward. The radius of a pro football is viii.5 cm at the heart of the short side. What is the centripetal acceleration of the laces on the football?

63.

A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal round path the riders follow has an 8.00-yard radius, at how many revolutions per minute are the riders subjected to a centripetal acceleration equal to that of gravity?

64 .

A runner taking role in the 200-grand dash must run around the finish of a track that has a circular arc with a radius of curvature of 30.0 one thousand. The runner starts the race at a abiding speed. If she completes the 200-g dash in 23.2 s and runs at constant speed throughout the race, what is her centripetal acceleration as she runs the curved portion of the rail?

65.

What is the acceleration of Venus toward the Sun, bold a round orbit?

66 .

An experimental jet rocket travels around Earth along its equator just to a higher place its surface. At what speed must the jet travel if the magnitude of its dispatch is g?

67.

A fan is rotating at a abiding 360.0 rev/min. What is the magnitude of the acceleration of a point on one of its blades 10.0 cm from the axis of rotation?

68 .

A indicate located on the 2d paw of a big clock has a radial dispatch of $\mathrm{0.i}\text{cm}\text{/}{\text{s}}^2.$ How far is the point from the centrality of rotation of the 2nd hand?

4.5 Relative Movement in One and Ii Dimensions

69.

The coordinate axes of the reference frame $S^{\prime }$ remain parallel to those of Southward, as ${\mathrm{Southward}}^{\prime }$ moves away from S at a abiding velocity ${\overrightarrow{v}}_{{\text{S}}^{\prime }}=(4.0\widehat{i}+3.0\widehat{j}+\mathrm{five.0}\widehat{k})\text{k}\text{/}\text{s}.$ (a) If at time t = 0 the origins coincide, what is the position of the origin $O^{\prime }$ in the S frame every bit a function of fourth dimension? (b) How is particle position for $\overrightarrow{r}(t)$ and ${\overrightarrow{r}}^{\prime }(t),$ as measured in South and ${\mathrm{Southward}}^{\prime },$ respectively, related? (c) What is the relationship between particle velocities $\overrightarrow{5}(t)\text{and}{\overrightarrow{5}}^{\prime }(t)?$ (d) How are accelerations $\overrightarrow{a}\text{(t)}\text{and}{\overrightarrow{a}}^{\prime }\text{(t)}$ related?

70 .

The coordinate axes of the reference frame $S^{\prime }$ remain parallel to those of S, every bit $S^{\prime }$ moves away from South at a constant velocity ${\overrightarrow{v}}_{S^{\prime }S}=(\mathrm{ane.0}\widehat{i}+\mathrm{two.0}\widehat{j}+3.0\widehat{\mathrm{thou}})t\text{m}\text{/}\text{s}$ . (a) If at time t = 0 the origins coincide, what is the position of origin $O^{\prime }$ in the South frame as a function of time? (b) How is particle position for $\overrightarrow{r}(t)$ and ${\overrightarrow{r}}^{\prime }(t)$ , as measured in S and ${\mathrm{Due\; south}}^{\prime },$ respectively, related? (c) What is the human relationship between particle velocities $\overrightarrow{v}(t)\text{and}{\overrightarrow{\mathrm{five}}}^{\prime }(t)?$ (d) How are accelerations $\overrightarrow{a}\text{(t) and}{\overrightarrow{a}}^{\prime }\text{(t)}$ related?

71.

The velocity of a particle in reference frame A is $(2.0\widehat{i}+3.0\widehat{j}\text{)}\text{m}\text{/}\text{southward}.$ The velocity of reference frame A with respect to reference frame B is $4.0\widehat{k}\text{m}\text{/}\text{s},$ and the velocity of reference frame B with respect to C is $2.0\widehat{j}\text{m}\text{/}\text{southward}.$ What is the velocity of the particle in reference frame C?

72 .

Raindrops fall vertically at 4.5 1000/s relative to the world. What does an observer in a auto moving at 22.0 m/s in a straight line measure out every bit the velocity of the raindrops?

73.

A seagull tin fly at a velocity of ix.00 m/s in all the same air. (a) If it takes the bird 20.0 min to travel 6.00 km straight into an oncoming wind, what is the velocity of the wind? (b) If the bird turns around and flies with the air current, how long will it take the bird to return half dozen.00 km?

74 .

A ship sets sail from Rotterdam, heading due northward at 7.00 chiliad/south relative to the water. The local ocean electric current is one.50 thousand/s in a direction $40.0\text{°}$ north of east. What is the velocity of the ship relative to Earth?

75.

A boat can exist rowed at 8.0 km/h in still h2o. (a) How much time is required to row one.v km downstream in a river moving iii.0 km/h relative to the shore? (b) How much fourth dimension is required for the return trip? (c) In what direction must the boat be aimed to row straight across the river? (d) Suppose the river is 0.8 km wide. What is the velocity of the boat with respect to Earth and how much fourth dimension is required to get to the reverse shore? (due east) Suppose, instead, the gunkhole is aimed straight across the river. How much fourth dimension is required to go beyond and how far downstream is the gunkhole when it reaches the opposite shore?

76 .

A minor plane flies at 200 km/h in still air. If the wind blows directly out of the westward at 50 km/h, (a) in what direction must the pilot head her plane to move direct due north across land and (b) how long does information technology take her to reach a bespeak 300 km directly north of her starting point?

77.

A cyclist traveling southeast along a road at 15 km/h feels a wind blowing from the southwest at 25 km/h. To a stationary observer, what are the speed and direction of the wind?

78 .

A river is moving e at four m/s. A gunkhole starts from the dock heading $30\text{°}$ due north of due west at 7 thou/s. If the river is 1800 m wide, (a) what is the velocity of the boat with respect to Earth and (b) how long does it take the boat to cross the river?

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