Two students perform an experiment with a train and a

Problem 1P Suppose that the component of a certain vector is doubled, (a) By what multiplicative factor docs the magnitude of the vector change? (b) By what multiplicative factor does the direction angle of the vector change?

CE Rank the vectors in Figure 3–31 in order of increasing magnitude. ________________ Equation Transcription: Text Transcription: vector{A} vector{D} vector{B} vector{C}

Given that \(\mathrm {\vec A + \vec B}=0\), (a) how does the magnitude of \(\mathrm {\vec B}\) compare with the magnitude of \(\mathrm {\vec A}\)? How does the direction of \(\mathrm {\vec B}\) compare with the direction of \(\mathrm {\vec A}\)? ________________ Equation Transcription: Text Transcription: vector{A}+vector{B}=0 vector{B} vector{A} vector{B} vector{A}

CE Rank the vectors in Figure 3–31 in order of increasing value of their x component. ________________ Equation Transcription: Text Transcription: vector{A} vector{D} vector{B} vector{C}

Problem 1CQ For the following quantities, indicate which is a scalar and which is a vector: (a) the time it takes for you to run the 100-yard dash; (b) your displacement after running the 100-yard dash; (c) your average velocity while running; (d) your average speed while running.

Problem 4CQ Can a component of a vector be greater than the vector’s magnitude?

Which, if any, of the vectors shown in Figure 3–30 are equal? ________________ Equation Transcription: Text Transcription: vector{A} vector{C} vector{B} vector{G} vector{H} vector{F} vector{D} vector{E} vector{I} vector{J} vector{K} vector{L}

Problem 84IP Referring to Example Suppose the boat has a speed of 6.7 m/s relative to the water, and that the dock on the opposite shore of the river is at the location x = 55 m and y = 28 m relative to the starting point of the boat, (a) At what angle relative to the x axis must the boat be pointed in order to reach the other dock? (b) With the angle found in part (a), what is the speed of the boat relative to the ground?

Problem 5P The press box at a baseball park is 32.0 ft above the ground. A reporter in the press box looks at an angle of 15.0° below the horizontal to see second base. What is the horizontal distance from the press box to second base?

CE Rank the vectors in Figure 3–31 in order of increasing value of their y component.

Problem 6CQ Can a vector with zero magnitude have one or more components that are nonzero? Explain.

Problem 6P You are driving up a long, inclined road. After 1.2 miles you notice that signs along the roadside indicate that your elevation has increased by 530 ft. (a) What is the angle of the road above the horizontal? (b) How far do you have to drive to gam an additional 150 ft of elevation?

Suppose that \(\mathrm {\vec A}\) and \(\mathrm {\vec B}\) have nonzero magnitude. Is it possible for \(\mathrm {\vec A}+\mathrm {\vec B}\) to be zero? ________________ Equation Transcription: Text Transcription: A B A+B

Problem 7P A One-Percent Grade A road that rises 1 ft for every 100 ft traveled horizontally is said to have a 1 % grade. Portions of the Lewiston grade, near Lewiston, Idaho, have a 6% grade. At what angle is this road inclined above the horizontal?

Given that \(\mathrm {\vec A}+\mathrm {\vec B}=\mathrm {\vec C}\), and that \(A-B=C\), how are \(\mathrm {\vec A}\) and \(\mathrm {\vec B}\) oriented relative to one another? ________________ Equation Transcription: Text Transcription: vector{A}+vector{B}=vector{C} vector{A}-vector{B}=vector{C} vector{A} vector{B}

Find the and components of a position vector \(\mathrm {\vec r}\) of magnitude \(r=75\ \mathrm m\), if its angle relative to the axis is (a) \(35.0^\circ\) and (b) \(65.0^\circ\). ________________ Equation Transcription: Text Transcription: vector{r} r=75 m 35.0^o 65.0^o

Given that \(\mathrm {\vec A}+\mathrm {\vec B}=\mathrm {\vec C}\), and that \(A^2+B^2=C^2\), how are \(\mathrm {\vec A}\) and \(\mathrm {\vec B}\) oriented relative to one another? ________________ Equation Transcription: Text Transcription: vector{A}+vector{B}=vector{C} A^2+B^2=C^2 vector{A} vector{B}

Given that \(\mathrm {\vec A}+\mathrm {\vec B}=\mathrm {\vec C}\), and that \(A+B=C\), how are \(\mathrm {\vec A}\) and \(\mathrm {\vec B}\) oriented relative to one another? ________________ Equation Transcription: Text Transcription: vector{A}+vector{B}=vector{C} A+B=C vector{A} vector{B}

Vector \(\mathrm {\vec A}\) has and components of equal magnitude. What can you say about the possible directions of \(\mathrm {\vec A}\)? ________________ Equation Transcription: Text Transcription: vector{A} vector{A}

The components of a vector \(\mathrm {\vec A}\) satisfy the relation \(A_x=-A_y\ne 0\). What are the possible directions of \(\mathrm {\vec A}\)? ________________ Equation Transcription: Text Transcription: vector{A} A_x=-A_y{not=}0 vector{A}

Alighthouse that rises 49 ft above the surface of the water sits on a rocky cliff that extends 19 ft from its base, as shown in Figure 3–33. A sailor on the deck of a ship sights the top of the lighthouse at an angle of \(30.0^\circ\) above the horizontal. If the sailor’s eye level is 14 ft above the water, how far is the ship from the rocks? ________________ Equation Transcription: Text Transcription: 30.0^o 30.0^o

IP The and components of a vector \(\mathrm {\vec r}\) are \(r_x=14\ \mathrm m\) and \(r_y=-9.5\ \mathrm m\), respectively. Find the direction and the magnitude of the vector \(\mathrm {\vec r}\). (c) If both \(r_x\) and \(r_y\) are doubled, how do your answers to parts (a) and (b) change? Equation Transcription: Text Transcription: vector{r} r_x=14 m r_y=-9.5 m vector{r} r_x r_y

Problem 12CQ Use a sketch to show that two vectors of unequal magnitude cannot add to zero, but that three vectors of unequal magnitude can.

\(\mathrm {H_2O}\) A water molecule is shown schematically in Figure 3-34. The distance from the center of the oxygen atom to the center of a hydrogen atom is \(0.96\ \AA\), and the angle between the hydrogen atoms is \(104.5^\circ\). Find the center-to-center distance between the hydrogen atoms. \(\mathrm{(1\ \AA=10^{-10}\ m.)}\) ________________ Equation Transcription: Å (1Å Å Text Transcription: H_2O 0.96 AA 104.5^o (1{AA}=10^-10 m.) 0.96 AA 104.5^o

IP The Longitude Problem In 1755 , John Harrison (1693-1776) completed his fourth precision chronometer, the H4, which eventually won the celebrated Longitude Prize. (For the human drama behind the Longitude Prize, see Longitude, by Dava Sobel.) When the minute hand of the indicated 10 minutes past the hour, it extended in the horizontal direction. (a) How long was the H4's minute hand? (b) At 10 minutes past the hour, was the extension of the minute hand in the vertical direction more than, less than, or equal to 3.0 cm? Explain. (c) Calculate the vertical extension of the minute hand at 10 minutes past the hour.

Problem 14CQ When sailing, the wind feels stronger when you sail upwind (“beating”) than when you are sailing downwind (“running”). Explain.

Problem 13CQ Rain is falling vertically downward and you are running for shelter. To keep driest, should you hold your umbrella vertically, tilted forward, or tilted backward? Explain.

Problem 14P You drive a car 680 ft to the east, then 340 ft to the north. (a) What is the magnitude of your displacement? (b) Using a sketch, estimate the direction of your displacement, (c) Verify your estimate in part (b) with a numerical calculation of the direction.

A baseball “diamond” (Figure 3–32) is a square with sides 90 ft in length. If the positive axis points from home plate to first base, and the positive axis points from home plate to third base, find the displacement vector of a base runner who has just hit (a) a double, (b) a triple, or (c) a home run.

A whale comes to the surface to breathe and then dives at an angle of \(20.0^\circ\) below the horizontal (Figure 3-35). If the whale continues in a straight line for , (a) how deep is it, and (b) how far has it traveled horizontally? ________________ Equation Transcription: Text Transcription: 20.0^o 20.0^o

Vector \(\mathrm {\vec A}\) has a magnitude of 50 units and points in the positive direction. A second vector, \(\mathrm {\vec B}\), has a magnitude of 120 units and points at an angle of \(70^\circ\) below the axis. Which vector has (a) the greater component, and the greater component? ________________ Equation Transcription: Text Transcription: vector{A} vector{B} 70^o

CE Refer to Figure 3-36 for the following questions: (a) Is the magnitude of \(\mathrm {\vec A + \vec D}\) greater than, less than, or equal to the magnitude of \(\mathrm {\vec A +\vec E}\)? (b) Is the magnitude of \(\mathrm {\vec A +\vec E}\) greater than, less than, or equal to the magnitude of \(\mathrm {\vec A +\vec F}\)? ________________ Equation Transcription: Text Transcription: vector{A}+vector{D} vector{A}+vector{E} vector{A}+vector{E} vector{A}+vector{F} vector{A} vector{B} vector{C} vector{D} vector{E} vector{F}

Problem 16P A treasure map directs you to start at a palm tree and walk due north for 15.0 m. You are then to turn 90° and walk 22.0 m; then turn 90° again and walk 5.00 m. Give the distance from the palm tree, and the direction relative to north, for the of the four possible locations of the treasure.

An air traffic controller observes two airplanes approaching the airport. The displacement from the control tower to plane 1 is given by the vector \(\mathrm {\vec A}\), which has a magnitude of and points in a direction \(32^\circ\) north of west. The displacement from the control tower to plane 2 is given by the vector \(\mathrm {\vec B}\), which has a magnitude of and points \(65^\circ\) east of north. (a) Sketch the vectors \(\mathrm {\vec A}\), \(-\mathrm {\vec B}\) and \(\mathrm {\vec D=\vec A-\vec B}\). Notice that \(\mathrm {\vec D}\) is the dis- placement from plane 2 to plane (b) Find the magnitude and direction of the vector \(\mathrm {\vec D}\). ________________ Equation Transcription: Text Transcription: vector{A} 32^o vector{B} vector{A} -vector{B} vector{D}=vector{A}-vector{B} vector{D} vector{D}

CE Consider the vectors \(\mathrm {\vec A}\) and \(\mathrm {\vec B}\) shown in Figure 3- 36 . Which of the other four vectors in the figure \(\mathrm{(\vec{C},\vec{D},\vec{E},\text{ and }\vec{F})}\) best represents the direction of (a) \(\mathrm {\vec{A} + \vec{B}}\), (b) \(\mathrm {\vec{A} - \vec{B}}\), and (c) \(\mathrm {\vec{B} - \vec{A}}\)? ________________ Equation Transcription: Text Transcription: vector{A} vector{B (vector{C},vector{D},vector{E}, and vector{F}) vector{A}+vector{B} vector{A}-vector{B} vector{B}-vector{A} vector{A} vector{B} vector{C} vector{D} vector{E} vector{F}

Vector \(\mathrm {\vec{A}}\) points in the negative direction and has a magnitude of 22 units. The vector \(\mathrm {\vec{B}}\) points in the positive direction. (a) Find the magnitude of \(\mathrm {\vec{B}}\) if \(\mathrm {\vec{A}+\vec{B}}\) has a magnitude of 37 units. (b) Sketch \(\mathrm {\vec{A}}\) and \(\mathrm {\vec{B}}\). ________________ Equation Transcription: Text Transcription: vector{A} vector{B} vector{B} vector{A}+vector{B} vector{A} vector{B}

A vector \(\mathrm {\vec{A}}\) has a magnitude of and points in a direction \(20.0^\circ\) below the positive axis. A second vector, \(\mathrm {\vec{B}}\), has a magnitude of and points in a direction \(50.0^\circ\) above the positive axis. (a) Sketch the vectors \(\mathrm {\vec A}\), \(\mathrm {\vec B}\) and \(\mathrm {\vec C=\vec A+\vec B}\). (b) Using the component method of vector addition, find the magnitude and direction of the vector \(\mathrm {\vec C}\). ________________ Equation Transcription: Text Transcription: vector{A} 20.0^o vector{B} 50.0^o vector{A} vector{B} vector{C}=vector{A}+vector{B} vector{C}

Vector \(\mathrm {\vec A}\) points in the positive direction and has a magnitude of . The vector \(\mathrm {\vec C=\vec A+\vec B}\) points in the positive direction and has a magnitude of . (a) Sketch \(\mathrm {\vec A}\), \(\mathrm {\vec B}\) and \(\mathrm {\vec C}\). (b) Estimate the magnitude and direction of the vector \mathrm {\vec B}.(c) Verify your estimate in part (b) with a numerical calculation. ________________ Equation Transcription: Text Transcription: vector{A} vector{C}=vector{A}+vector{B} vector{A} vector{B} vector{C} vector{B}

A particle undergoes a displacement \(\Delta \mathrm {\vec r}\) of magnitude in a direction \(42^\circ\) below the axis. Express \(\Delta \mathrm {\vec r}\) in terms of the unit vectors \(\hat {\mathrm x}\) and \(\hat {\mathrm y}\). ________________ Equation Transcription: Text Transcription: {Delta}vector{r} 42^o {Delta}vector{r} hat{x} hat{y}

A basketball player runs down the court, following the path indicated by the vectors \(\mathrm {\vec A}\), \(\mathrm {\vec B}\) and \(\mathrm {\vec C}\) in Figure . The magnitudes of these three vectors are \(A=10.0\ \mathrm m\), \(B=20.0\ \mathrm m\) and \(C=7.0\ \mathrm m\). Find the magnitude and direction of the net displacement of the player using (a) the graphical method and (b) the component method of vector addition. Compare your results. ________________ Equation Transcription: Text Transcription: vector{A} vector{B} vector{C} A=10.0 m B=20.0 m C=7.0 m vector{A} 45^o vector{B} vector{C} 30^o

Problem 28P A vector has a magnitude of 3.50 m and points in a direction that is 145° counterclockwise from the x axis. Find the x and y components of this vector.

Vector \(\mathrm {\vec A}\) points in the negative direction and has a magnitude of 5 units. Vector \(\mathrm {\vec B}\) has twice the magnitude and points in the positive direction. Find the direction and magnitude of (a) \(\mathrm {\vec A + \vec B}\), \(\mathrm {\vec A - \vec B}\) and (c) \(\mathrm {\vec B - \vec A}\). ________________ Equation Transcription: Text Transcription: vector{A} vector{B} vector{A}+vector{B} vector{A}-vector{B} vector{B}-vector{A}

The initial velocity of a car, \(\mathrm {\vec v _i}\), is in the positive direction. The final velocity of the car, \(\mathrm {\vec v _f}\), is in a direction that points \(75^\circ\) above the positive axis. (a) Sketch the vectors \(-\mathrm {\vec v _i}\), \(\mathrm {\vec v _f}\) and \(\Delta \mathrm {\vec v= \vec A_f-\vec A_i}\). (b) Find the magnitude and direction of the change in velocity, \(\Delta \mathrm {\vec v}\). ________________ Equation Transcription: Text Transcription: vector{v}_i vector{v}_f 75^o -vector{v}_i vector{v}_f {Delta}vector{v}=vector{A}_f-vector{A}_i {Delta}vector{v}

The vector \(-5.2\ \mathrm {\vec A}\) has a magnitude of and points in the positive direction. Find the component and the magnitude of the vector \(\mathrm {\vec A}\). ________________ Equation Transcription: Text Transcription: -5.2 vector{A} vector{A}

A vector \(\mathrm {\vec A}\) has a length of and points in the negative direction. Find the component and (b) the magnitude of the vector \(-3.7\ \mathrm {\vec A}\). ________________ Equation Transcription: Text Transcription: vector{A} -3.7 vector{A}

Find the direction and magnitude of the vectors. (a) \(\mathrm {\vec{A}=(5.0\ m)\hat{x}+(-2.0\ m)\hat{y}}\) (b) \(\mathrm {\vec{B}=(-2.0\ m)\hat{x}+(5.0\ m)\hat{y}}\), and (c) \(\mathrm {\vec{A}+\vec{B}}\). ________________ Equation Transcription: Text Transcription: vector{A}=(5.0 m)hat{x}+(-2.0 m)hat{y} vector{B}=(-2.0 m)hat{x}+(5.0 m)hat{y} vector{A}+vector{B}

Referring to the vectors in Figure 3-38, express the sum \(\mathrm {\vec{A}+\vec{B}+\vec{C}}\) in unit vector notation. ________________ Equation Transcription: Text Transcription: vec{A}+vec{B}+vec{C} vec{A} 40^o vec{C} 25^o 19^o vec{B}

For the vectors given in Problem 32, express (a) \(\mathrm {\vec{A}-\vec{B}}\) and (b) \(\mathrm {\vec{B}-\vec{A}}\) in unit vector notation. ________________ Equation Transcription: Text Transcription: vec{A}-vec{B} vec{B}-vec{A}

Find the direction and magnitude of the vectors. (a) \(\mathrm {\vec{A}=(25\ m)\hat x+(-12\ m)\hat y}\) (b) \(\mathrm {\vec{B}=(2.0\ m)\hat x+(15\ m)\hat y}\), and (c) \(\). Equation Transcription: Text Transcription: vec{A}=(25 m)hat{x} +(-12 m)hat{y} vec{B}=(2.0 m)hat{x}+(15 m)hat{y} vec{A}+vec{B}

Express each of the vectors in Figure 3-38 in unit vector notation. ________________ Equation Transcription: Text Transcription: vec{A} vec{B} vec{C} vec{D} 40^o 25^o 19^o

Problem 40P What are the direction and magnitude of your total displacement if you have traveled due west with a speed of 27 m/s for 125 s, their due south at 14 m/s for 66 s?

CE The blue curves shown in Figure 3–39 display the constant speed motion of two different particles in the ????-???? plane. For each of the eight vectors in Figure 3–39, state whether it is (a) a position vector, (b) a velocity vector, or (c) an acceleration vector for the particles.

Problem 38P In its daily prowl of the neighborhood, a cat makes a displacement of 120 m due north, followed by a 72-m displacement due west, (a) Find the magnitude and direction of the displacement required for the cat to return home, (b) If, instead, the cat had first prowled 72 m west and then 120 m north, how would this affect the displacement needed to bring it home? Explain.

Problem 39P If the cat in Problem takes 45 minutes to complete the 120-m displacement and 17 minutes to complete the 72-m displacement, what arc the magnitude and direction of its average velocity during this 62-minute period of time? In its daily prowl of the neighborhood, a cat makes a displacement of 120 m due north, followed by a 72-m displacement due west, (a) Find the magnitude and direction of the displacement required for the cat to return home, (b) If, instead, the cat had first prowled 72 m west and then 120 m north, how would this affect the displacement needed to bring it home? Explain.

You drive a car 1500 ft to the east, then 2500 ft to the north. If the trip took 3.0 minutes, what were the direction and magnitude of your average velocity?

IP Moving the Knight Two of the allowed chess moves for a knight are shown in Figure 3–40. (a) Is the magnitude of displacement 1 greater than, less than, or equal to the magnitude of displacement 2? Explain. (b) Find the magnitude and direction of the knight’s displacement for each of the two moves. Assume that the checkerboard squares are 3.5 cm on a side.

A jogger runs with a speed of 3.25 m/s in a direction 30.0° above the x axis, (a) Find the x and y components of the jogger’s velocity, (b) How will the velocity components found in part (a) change if the jogger’s speed is halved?

IP The Position of the Moon Relative to the center of the Earth, the position of the Moon can be approximated by \(\vec{r}=\left(3.84 \times 10^{8} \mathrm{~m}\right)\left\{\cos \left[\left(2.46 \times 10^{-6} \mathrm{radians} / \mathrm{s}\right) t\right] \widehat{x}+\sin \left[\left(2.46 \times 10^{-6} \mathrm{radians} / \mathrm{s}\right) t\right] \hat{y}\right\}\) where \(t\) is measured in seconds. (a) Find the magnitude and direction of the Moon's average velocity between \(t=0\) and \(t=7.38\) days. (This time is one-quarter of the 29.5 days it takes the Moon to complete one orbit.) (b) Is the instantaneous speed of the Moon greater than, less than, or the same as the average speed found in part (a)? Explain. Equation Transcription: Text Transcription: vec r=(3.84x10^8m) {cos[(2.4610^-6radians/s)t]widehat x +sin[(2.4610^-6radians/s)t]}widehat y t t=0 t=7.38

Problem 44P A skateboarder rolls from rest down an inclined ramp that is 15.0 m long and inclined above the horizontal at an angle of ? = 20.0°. When she reaches the bottom of the ramp 3.00 s later her speed is 10.0 m/s. Show that the average acceleration of the skateboarder is g sin ?, where g = 9.81 m/s2.

The Velocity of the Moon The velocity of the Moon relative to the center of the Earth can be approximated by \(\vec{v}=(945 \mathrm{~m} / \mathrm{s})\left\{-\sin \left[\left(2.4 \times 10^{-6} \mathrm{radians} / \mathrm{s}\right) t\right] \hat{x}+\cos \left[\left(2.46 \times 10^{-6} \mathrm{radians} / \mathrm{s}\right) t\right] \mathcal{I}\right\}\) where t is measured in seconds. To approximate the instantaneous acceleration of the Moon at t=0, calculate the magnitude and direction of the average acceleration between the times (a) t=0 and t=0.100 days and (b) t=0 and days. (The time required for the Moon to complete one orbit is 29.5 days.) Equation Transcription: Text Transcription: vec v=(945 m/s){-sin[ (2.410-6radians/s)t]hat x+cos[( 2.4610-6radians/s)t]hat y} t=0 t=0 t=0.100 t=0 t=0.0100

Problem 45P Consider a skateboarder who starts from rest at the top of a ramp that is inclined at an angle of 17.5° to the horizontal. Assuming that the skateboarder’s acceleration is g sin 17.5°, Find his speed when he reaches the bottom of the ramp in 3.25 s.

Problem 43P You throw a ball upward with an initial speed of 4.5 m/s. When it returns to your hand 0.92 s later, it has the same speed in the downward direction (assuming air resistance can be ignored). What was the average acceleration vector of the ball?

CE The accompanying photo shows a KC-10A Extender using a boom to refuel an aircraft in flight. If the velocity of the is due east relative to the ground, what is the velocity of the aircraft being refueled relative to (a) the ground, and (b) the KC-10A?

Problem 50P Referring to part (a) of Example, find the time it takes for the boat to reach the opposite shore if the river is 35 m wide.

Problem 51P As you hurry to catch your flight at the local airport, you encounter a moving walkway that is 85 m long and has a speed of 2.2 m/s relative to the ground. If it takes you 68 s to cover 85 m when walking on the ground, how long will it take you to cover the same distance on the walkway? Assume that you walk with the same speed on the walkway as you do on the ground.

Problem 52P In Problem, how long would it take you to cover the 85-m length of the walkway if, once you get on the walkway, you immediately turn around and start walking in the opposite direction with a speed of 1.3 m/s relative to the walkway? As you hurry to catch your flight at the local airport, you encounter a moving walkway that is 85 m long and has a speed of 2.2 m/s relative to the ground. If it takes you 68 s to cover 85 m when walking on the ground, how long will it take you to cover the same distance on the walkway? Assume that you walk with the same speed on the walkway as you do on the ground.

Problem 54P A passenger walks from one side of a ferry to the other as it approaches a dock. If the passenger’s velocity is 1.50 m/s due north relative to the ferry, and 4.50 m/s at an angle of 30.0° west of north relative to the water, what are the direction and magnitude of the Ferry’s velocity relative to the water?

Problem 56P In Problem, suppose the Jet Ski is moving at a speed of 12 m/s relative to the water, (a) At what angle must you point the Jet Ski if your velocity relative to the ground is to be perpendicular to the shore of the river? (b) If you increase the speed of the Jet Ski relative to the water, does the angle in part (a) increase, decrease, or stay the same? Explain. (Note: Angles are measured relative to the x axis shown, in Example.) You are riding on a Jet Ski at an angle of 35° upstream, on a river flowing with a speed of 2.8 m/s. If your velocity relative to the ground is 9.5 m/s at an angle of 20.0° upstream, what is the speed of the Jet Ski relative to the water? (Note: Angles are measured relative to the x axis shown in Example.)

Problem 53P The pilot of an airplane wishes to fly due north, but there is a 65-km/h wind blowing toward the east, (a) In what direction should the pilot head her plane if its speed relative to the air is 340 km/h? (b) Draw a vector diagram that illustrates your result in part (a), (c) If the pilot decreases the air speed of the plane, but still wants to head due north, should the angle found in part (a) be increased or decreased?

Problem 49P As an airplane taxies on the runway with a speed of 16.5 m/s, a Flight attendant walks toward the tail of the plane with a speed of 1.22 m/s. What is the flight attendant’s speed relative to the ground?

CE Predict/Explain Two vectors are defined as follows: \(\mathrm {\vec{A}=(-2.2\ m)\hat x}\) and \(\mathrm {\vec{B}=(1.4\ m)\hat y}\). (a) Is the magnitude of \(1.4\ \mathrm {\vec{A}}\) greater than, less than, or equal to the magnitude of \(2.2\ \mathrm {\vec{B}}\)? (b) Choose the best explanation from among the following: I. The vector \(\mathrm {\vec{A}}\) has a negative component. II. A number and its negative have the same magnitude. III. The vectors \(1.4\ \mathrm {\vec{A}}\) and \(2.2\ \mathrm {\vec{B}}\) point in opposite directions. ________________ Equation Transcription: Text Transcription: vec{A}=(-2.2 m)hat{x} vec{B}=(1.4 m)hat{y} 1.4 vec{A} 2.2 vec{B} vec{A} 1.4 vec{A} 2.2 vec{B}

Problem 57P Two people take identical Jet Skis across a river, traveling at the same speed relative to the water. Jet Ski A heads directly across the river and is carried downstream by the current before teaching the opposite shore. Jet Ski B travels in a direction that is 35° upstream and arrives at the opposite shore directly across from the starting point, (a) Which Jet Ski reaches the opposite shore in the least amount of time? (b) Confirm your answer to part (a) by finding the ratio of the time it takes for the two Jet Skis to cross the river. (Note: Angles are measured relative to the x axis shown in Example.)

CE The components of a vector \(\mathrm {\vec{B}}\) satisfy \(B_x>0\) and \(B_y<0\). Is the direction angle of \(\mathrm {\vec{B}}\) between \(0^\circ\) and \(90^\circ\), between \(90^\circ\) and \(180^\circ\), between \(180^\circ\) and \(270^\circ\), or between \(270^\circ\) and \(360^\circ\)? ________________ Equation Transcription: Text Transcription: vec{B} B_x>0 B_y<0 vec{B} 0^o 90^o 90^o 180^o 180^o 270^o 270^o 360^o

It is given that \(\mathrm {\vec{A}-\vec{B}=(-51.4\ m)\hat{x}}\), \(\mathrm {\vec{C}=(62.2\ m)\hat{x}}\) and \(\mathrm {\vec{A}+\vec{B}+\vec{C}=(13.8\ m)\hat{x}}\). Find the vectors \(\mathrm {\vec{A}}\) and \(\mathrm {\vec{B}}\). ________________ Equation Transcription: Text Transcription: vec{A}-vec{B}=(-51.4 m)hat{x} vec{C}=(62.2 m)hat{x} vec{A}+vec{B}+vec{C}=(13.8 m)hat{x} vec{A} vec{B}

CE The components of a vector \(\mathrm {\vec{A}}\) satisfy \(A_x<0\) and \(A_y<0\). Is the direction angle of \(\mathrm {\vec{A}}\) between \(0^\circ\) and \(90^\circ\), between \(90^\circ\) and \(180^\circ\), between \(180^\circ\) and \(270^\circ\), or between \(270^\circ\) and \(360^\circ\)? ________________ Equation Transcription: Text Transcription: vec{A} A_x<0 A_y<0 vec{A} 0^o 90^o 90^o 180^o 180^o 270^o 270^o 360^o

Problem 60GP You slide a box up a loading ramp that is 10.0 ft long. At the top of the ramp the box has risen a height of 3.00 ft. What is the angle of the ramp above the horizontal?

Find the , and components of the vector \(\mathrm {\vec{A}}\) shown in Figure 3-41, given that \(A=65\ \mathrm m\). ________________ Equation Transcription: Text Transcription: vec{A} A=65 m 55^o vec{A} 35^o

Find the direction and magnitude of the vector \(2\mathrm {\vec{A}+\vec{B}}\), where \(\mathrm {\vec{A}=(12.1\ m)\hat x}\) and \(\mathrm {\vec{B}=(-32.2\ m)\hat y}\). ________________ Equation Transcription: Text Transcription: 2vec{A}+vec{B} vec{A}=(12.1 m)hat{x} vec{B}=(-32.2 m)y

Problem 66GP An off-roader explores the open desert in her Hummer. First she drives 25° west of north with a speed of 6.5 km/h for 15 minutes, then due east with a speed of 12 km/h for 7.5 minutes. She completes the final leg of her trip in 22 minutes. What are the direction and speed of travel on the final leg? (Assume her speed is constant on the leg, and that she returns to her starting point at the end of the final leg.)

CE Predict/Explain Consider the vectors \(\mathrm {\vec{A}=(1.2\ m)\hat{x}}\) and \(\mathrm {\vec{B}=(-3.4\ m)\hat{x}}\). (a) Is the magnitude of vector \(\mathrm {\vec{A}}\) greater than, less than, or equal to the magnitude of vector \(\mathrm {\vec{B}}\) ? (b) Choose the best explanation from among the following: I. The number is greater than the number . II. The component of \(\mathrm {\vec{B}}\) is negative. III. The vector \(\mathrm {\vec{A}}\) points in the positive direction. ________________ Equation Transcription: Text Transcription: vec{A}=(1.2 m)hat{x} vec{B}=(-3.4 m)hat{x} vec{A} vec{B} vec{B} vec{A}

Problem 65GP Two students perform an experiment with a train and a ball. Michelle tides on a flatcar pulled at 8.35 m/s by a train on a straight, horizontal track; Gary stands at rest on the ground near the tracks. When Michelle throws the ball with an initial angle of 65.0° above the horizontal, from her point of view, Gary sees the ball rise straight up and back down above a fixed point on the ground, (a) Did Michelle throw the ball toward the front of the train or toward the rear of the train? Explain, (b) What was the initial speed of Michelle’s throw? (c) What was the initial speed of the ball as seen by Gary?

As a function of time, the velocity of the football described in Problem 68 can be written as \(\mathrm {\vec{v}=(16.6\ m/s)\hat{x}-[(9.81\ m/s^2)}t]\mathrm {\hat{y}}\). Calculate the average acceleration vector of the football for the time periods (a) \(t=0\) to \(t=1.00\ \mathrm s\), (b) \(t=0\) to \(t=2.50\ \mathrm s\), and (c) \(t=0\) to \(t=5.00\ \mathrm s\). (If the acceleration of an object is constant, its average acceleration is the same for all time periods.) ________________ Equation Transcription: Text Transcription: vec{v}=(16.6 m/s)hat{x}-[(9.81 m/s^2)t]hat{y} t=0 t=1.00 s t=0 t=2.50 s t=0 t=5.00 s

Two airplanes taxi as they approach the terminal. Plane 1 taxies with a speed of \(12 \mathrm{~m} / \mathrm{s}\) due north. Plane 2 taxies with a speed of \(7.5 \mathrm{~m} / \mathrm{s}\) in a direction \(20^{\circ}\) north of west. (a) What are the direction and magnitude of the velocity of plane 1 relative to plane 2? (b) What are the direction and magnitude of the velocity of plane 2 relative to plane 1 ?

A football is thrown horizontally with an initial velocity of \(\mathrm {(16.6\ m/s)\hat{x}}\). Ignoring air resistance, the average accelerationof the football over any period of time is \(\mathrm {(-9.81\ m/s^2)\hat{y}}\). (a) Find the velocity vector of the ball 1.75 s after it is thrown. (b) Find the magnitude and direction of the velocity at this time. ________________ Equation Transcription: Text Transcription: (16.6 m/s)hat{x} (-9.81 m/s^2)hat{y}

Problem 72GP Initially, a particle is moving at 4.10 m/s at an angle of 33.5° above the horizontal. Two seconds later, its velocity is 6.05 m/s at an angle of 59.0° below the horizontal. What was the particle’s average acceleration during these 2.00 seconds?

Problem 74GP A Big Clock The clock that rings the bell known as Big Ben has an hour hand that is 9.0 feet long and a minute hand that is 14 feet long, where the distance is measured from the center of the clock to the tip of the hand. What is the tip-to-tip distance between these two hands when the clock reads 12 minutes after four o’clock?

A shopper at the supermarket follows the path indicated by vectors \(\mathrm {\vec A,\vec B,\vec C}\), and \(\mathrm {\vec D}\) in Figure Given that the vectors have the magnitudes \(A=51\ \mathrm {ft}\), \(B=45\ \mathrm {ft}\), \(C=35\ \mathrm {ft}\) and \(D=13\ \mathrm {ft}\), find the total displacement of the shopper using (a) the graphical method and (b) the component method of vector addition. Give the direction of the displacement relative to the direction of vector \(\mathrm {\vec A}\). Equation Transcription: Text Transcription: vec{A},vec{B},vec{C} vec{D} vec{A} vec{B} vec{C} vec{D} A=51 ft B=45 ft C=35 ft D=13 ft

IP Suppose we orient the axis of a two-dimensional coordinate system along the beach at Waikiki. Waves approaching the beach have a velocity relative to the shore given by \(\mathrm {\vec v_{ws}=(1.3\ m/s)\hat y}\). Surfers move more rapidly than the waves, but at an angle to the beach. The angle is chosen so that the surfers approach the shore with the same speed as the waves. (a) If a surfer has a speed of relative to the water, what is her direction of motion relative to the positive axis? (b) What is the surfer's velocity relative to the wave? (c) If the surfer's speed is increased, will the angle in part (a) increase or decrease? Explain. ________________ Equation Transcription: Text Transcription: vec{v}_ws=(1.3 m/s)y

Problem 73GP A passenger on a stopped bus notices that rain is falling vertically just outside the window. When the bus moves with constant velocity, the passenger observes that the falling raindrops are now making an angle of 15° with respect to the vertical. (a) What is the ratio of the speed of the raindrops to the speed of the bus? (b) Find the speed of the raindrops, given that the bus is moving with a speed of 18m/s.

Problem 78P As two boats approach the marina, the velocity of boat 1 relative to boat 2 is 2.15 m/s in a direction 47.0° east of north. If boat 1 has a velocity that is 0.775 m/s due north, what is the velocity (magnitude and direction) of boat 2?

Vector \(\mathrm {\vec A}\) points in the negative direction. Vector \(\mathrm {\vec B}\) points at an angle of \(30.0^\circ\) above the positive axis. Vector \(\mathrm {\vec C}\) has a magnitude of and points in a direction \(40.0^\circ\) below the positive axis. Given that \(\mathrm {\vec A+\vec B+\vec C=0}\), find the magnitudes of \(\mathrm {\vec A}\) and \(\mathrm {\vec B}\). ________________ Equation Transcription: Text Transcription: vec{A} vec{B} 30.0^o vec{C} 40.0^o vec{A}+vec{B}+vec{C}=0 vec{A} vec{B}

IP Referring to Example , (a) what heading must the boat have if it is to land directly across the river from its starting point? (b) How much time is required for this trip if the river is wide? (c) Suppose the speed of the boat is increased, but it is still desired to land directly across from the starting point. Should the boat's heading be more upstream, more downstream, or the same as in part (a)? Explain.

Problem 79PP What speed must the dragonfly have if the line of sight, which is parallel to the ?x axis initially, is to remain parallel to the ?x? axis? A?. 0.562 m/s B?. 0.664 m/s C?. 1.00 m/s D?. 1.13 m/s FIGURE

If the dragonfly approaches its prey with a speed of , what angle \(\theta\) is required to maintain a constant line of sight parallel to the axis? A. \(37.9^\circ\) B. \(38.3^\circ\) C. \(51.7^\circ\) D. \(52.1^\circ\) ________________ Equation Transcription: Text Transcription: theta 37.9o 38.3o 51.7o 52.1o

Suppose the dragonfly now approaches its prey along a path with \(\theta > 48.5^\circ\), but it still keeps the line of sight parallel to the axis. Is the speed of the dragonfly in this new case greater than, less than, or equal to its speed in Problem 79 ? ________________ Equation Transcription: Text Transcription: theta>48.5^o

What is the correct "motion camouflage" speed of approach for a dragonfly pursuing its prey at the angle \(\theta = 68.5^\circ\)? A. B. C. D. ________________ Equation Transcription: Text Transcription: theta=68.5^o

Problem 83IP Referring to Example Suppose the speed of the boat relative to the water is 7.0 m/s. (a) At what angle to the x axis must the boat be headed if it is to land directly across the river from its starting position? (b) If the speed of the boat relative to the water is increased, will the angle needed to go directly across the river increase, decrease, or stay the same? Explain.

Problem 55P You are riding on a Jet Ski at an angle of 35° upstream, on a river flowing with a speed of 2.8 m/s. If your velocity relative to the ground is 9.5 m/s at an angle of 20.0° upstream, what is the speed of the Jet Ski relative to the water? (Note: Angles are measured relative to the x axis shown in Example.)

What speed must the dragonfly have if the line of sight, which is parallel to the x axis initially, is to remain parallel to the x axis? A. \(0.562 \mathrm{~m} / \mathrm{s}\) B. \(0.664 \mathrm{~m} / \mathrm{s}\) C. \(1.00 \mathrm{~m} / \mathrm{s}\) D. \(1.13 \mathrm{~m} / \mathrm{s}\) Equation Transcription: Text Transcription: 0.562 m/s 0.664 m/s 1.00 m/s 1.13 m/s