Solved: You drive a car 680 ft to the east, then 340 ft to

Suppose that each component of a certain vector is doubled. (a) By what multiplicative factor does the magnitude of the vector change? (b) By what multiplicative factor does the direction angle of the vector change?

Rank the vectors in Figure 331 in order of increasing magnitude.

Rank the vectors in Figure 331 in order of increasing value of their x component.

Rank the vectors in Figure 331 in order of increasing value of their y component.

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?

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 gain an additional 150 ft of elevation?

A One-Percent Grade Aroad 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?

Find the x and y components of a position vector of magnitude , if its angle relative to the x axis is (a) 35.0 and (b) 65.0.

Abaseball diamond (Figure 332) is a square with sides 90 ft in length. If the positive x axis points from home plate to rst base, and the positive y axis points from home plate to third base, nd the displacement vector of a base runner who has just hit (a) a double, (b) a triple, or (c) a home run.

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 333. A sailor on the deck of a ship sights the top of the lighthouse at an angle of 30.0above the horizontal. If the sailors eye level is 14 ft above the water, how far is the ship from the rocks?

H2OAwater molecule is shown schematically in Figure 334. The distance from the center of the oxygen atom to the center of a hydrogen atom is 0.96 , and the angle between the hydrogen atoms is 104.5. Find the center-to-center distance between the hydrogen atoms. (1 = 10-10 m.)

The x and y components of a vector are and respectively. Find (a) the direction and (b) the magnitude of the vector (c)If both and are doubled, how do your answers to parts (a) and (b) change?

The Longitude Problem In 1755, John Harrison (16931776) 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 H4 indicated 10 minutes past the hour, it extended 3.0 cm in the horizontal direction. (a) How long was the H4s 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.

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.

Vector has a magnitude of 50 units and points in the positive x direction. A second vector, , has a magnitude of 120 units and points at an angle of 70 below the x axis. Which vector has (a) the greater x component, and (b) the greater y component?

Atreasure 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 90again and walk 5.00 m. Give the distance from the palm tree, and the direction relative to north, for each of the four possible locations of the treasure.

Awhale comes to the surface to breathe and then dives at an angle of 20.0 below the horizontal (Figure 335). If the whale continues in a straight line for 150 m, (a) how deep is it, and (b) how far has it traveled horizontally?

Consider the vectors and shown in Figure 336. Which of the other four vectors in the gure and best represents the directionof(a) (b) and(c) ?

Refer to Figure 336 for the following questions: (a) Is the magnitude of greater than, less than, or equal to the magnitude of ? (b) Is the magnitude of greater than, less than, or equal to the magnitude of ?

A vector has a magnitude of 40.0 m and points in a direction 20.0 below the positive x axis. A second vector, , has a magnitude of 75.0 m and points in a direction 50.0 above the positive x axis. (a) Sketch the vectors , and . (b) Using the component method of vector addition, nd the magnitude and direction of the vector .

An air trafc controller observes two airplanes approaching the airport. The displacement from the control tower to plane 1 is given by the vector , which has a magnitude of 220 km and points in a direction 32 north of west. The displacement from the control tower to plane 2 is given by the vector , which has a magnitude of 140 km and points 65 east of north. (a) Sketch the vectors , and . Notice that is the displacement from plane 2 to plane 1. (b) Find the magnitude and direction of the vector .

The initial velocity of a car, , is 45 km/h in the positive xdirection. The nal velocity of the car, , is 66 km/h in a direction that points 75 above the positive x axis. (a) Sketch the vectors , and . (b) Find the magnitude and direction of the change in velocity, .

Vector points in the positive x direction and has a magnitude of 75 m. The vector points in the positive y direction and has a magnitude of 95 m. (a) Sketch , and . (b) Estimate the magnitude and direction of the vector . (c)Verify your estimate in part (b) with a numerical calculation.

Vector points in the negative x direction and has a magnitude of 22 units. The vector points in the positive y direction. (a) Find the magnitude of if has a magnitude of 37 units. (b) Sketch and .

Vector points in the negative ydirection and has a magnitude of 5 units. Vector has twice the magnitude and points in the positive x direction. Find the direction and magnitude of (a) , (b) , and (c) .

A basketball player runs down the court, following the path indicated by the vectors , and in Figure 337. The magnitudes of these three vectors are and . 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.

Aparticle undergoes a displacement of magnitude 54 m in a direction 42 below the x axis. Express in terms of the unit vectors

Avector 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.

Avector has a length of 6.1 m and points in the negative x direction. Find (a) the x component and (b) the magnitude of the vector

The vector has a magnitude of 34 m and points in the positive x direction. Find (a) the x component and (b) the magnitude of the vector

Find the direction and magnitude of the vectors. (a) , (b) , and (c) .

Find the direction and magnitude of the vectors. (a) , (b) , and (c) .

For the vectors given in Problem 32, express (a) and (b) in unit vector notation.

Express each of the vectors in Figure 338in unit vector notation.

Referring to the vectors in Figure 338, express the sum in unit vector notation

The blue curves shown in Figure 339display the constantspeed motion of two different particles in the x-y plane. For each of the eight vectors in Figure 339, state whether it is (a) a position vector, (b) a velocity vector, or (c) an acceleration vector for the particles.

Moving the Knight Two of the allowed chess moves for a knight are shown in Figure 340. (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 knights displacement for each of the two moves. Assume that the checkerboard squares are 3.5 cm on a side.

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 rst prowled 72 m west and then 120 m north, how would this affect the displacement needed to bring it home? Explain.

If the cat in Problem 38 takes 45 minutes to complete the 120-m displacement and 17 minutes to complete the 72-m displacement, what are the magnitude and direction of its average velocity during this 62-minute period of time?

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, then due south at 14 m/s for 66 s?

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?

Ajogger runs with a speed of 3.25 m/s in a direction 30.0 above the xaxis.(a)Find the xandycomponents of the joggers velocity. (b)How will the velocity components found in part (a) change if the joggers speed is halved?

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?

Askateboarder rolls from rest down an inclined ramp that is 15.0 m long and inclined above the horizontal at an angle of . 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 , where .

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 skateboarders acceleration is g sin 17.5, nd his speed when he reaches the bottom of the ramp in 3.25 s.

The Position of the Moon Relative to the center of the Earth, the position of the Moon can be approximated by where tis measured in seconds. (a)Find the magnitude and direction of the Moons average velocity between and 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.

The Velocity of the Moon The velocity of the Moon relative to the center of the Earth can be approximated by where t is measured in seconds. To approximate the instantaneous acceleration of the Moon at , calculate the magnitude and direction of the average acceleration between the times (a) and days and (b) and days. (The time required for the Moon to complete one orbit is 29.5 days.)

The accompanying photo shows a KC-10AExtender using a boom to refuel an aircraft in ight. If the velocity of the KC10Ais 125 m/s 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?

As an airplane taxies on the runway with a speed of 16.5 m/s, a ight attendant walks toward the tail of the plane with a speed of 1.22 m/s. What is the ight attendants speed relative to the ground?

Referring to part (a) of Example 32, nd the time it takes for the boat to reach the opposite shore if the river is 35 m wide.

As you hurry to catch your ight 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.

In Problem 51, 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?

The pilot of an airplane wishes to y 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?

Apassenger walks from one side of a ferry to the other as it approaches a dock. If the passengers velocity is 1.50 m/s due north relative to the ferry, and 4.50 m/s at an angle of 30.0west of north relative to the water, what are the direction and magnitude of the ferrys velocity relative to the water?

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 32.)

In Problem 55, 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 32.)

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 reaching 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) Conrm your answer to part (a) by nding 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 32.)

Predict/Explain Consider the vectors and . (a) Is the magnitude of vector greater than, less than, or equal to the magnitude of vector ? (b)Choose the best explanation from among the following: I. The number 3.4 is greater than the number 1.2. II. The component of is negative. III. The vector points in the positive x direction.

Predict/Explain Two vectors are dened as follows: and . (a) Is the magnitude of 1.4 greater than, less than, or equal to the magnitude of 2.2 ? (b) Choose the best explanation from among the following: I. The vector has a negative component. II. Anumber and its negative have the same magnitude. III. The vectors 1.4 and 2.2 point in opposite directions.

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 direction and magnitude of the vector , where and .

The components of a vector satisfy and . Is the direction angle of between 0and 90, between 90and 180, between 180and 270, or between 270and 360?

The components of a vector satisfy and . Is the direction angle of between 0and 90, between 90and 180, between 180 and 270, or between 270 and 360?

It is given that , and . Find the vectors and .

Two students perform an experiment with a train and a ball. Michelle rides on a atcar 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 xed 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 Michelles throw? (c)What was the initial speed of the ball as seen by Gary?

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 nal leg of her trip in 22 minutes. What are the direction and speed of travel on the nal leg? (Assume her speed is constant on each leg, and that she returns to her starting point at the end of the nal leg.)

Find the x, y, and z components of the vector shown in Figure 341, given that .

A football is thrown horizontally with an initial velocity of . Ignoring air resistance, the average acceleration of the football over any period of time is . (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.

As a function of time, the velocity of the football described in Problem 68 can be written as . Calculate the average acceleration vector of the football for the time periods (a) to , (b) to , and (c) to . (If the acceleration of an object is constant, its average acceleration is the same for all time periods.)

Two airplanes taxi as they approach the terminal. Plane 1 taxies with a speed of 12 m/s due north. Plane 2 taxies with a speed of 7.5 m/s in a direction 20 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 shopper at the supermarket follows the path indicated by vectors , and in Figure 342. Given that the vectors have the magnitudes , and nd 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 .

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 particles average acceleration during these 2.00 seconds?

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 15with 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 18 m/s.

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 each hand. What is the tip-to-tip distance between these two hands when the clock reads 12 minutes after four oclock?

Suppose we orient the x 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 . 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 7.2 m/s relative to the water, what is her direction of motion relative to the positive x axis? (b) What is the surfers velocity relative to the wave? (c) If the surfers speed is increased, will the angle in part (a) increase or decrease? Explain.

Referring to Example 32, (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 25.0 m 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 boats heading be more upstream, more downstream, or the same as in part (a)? Explain.

Referring to Example 32, (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 25.0 m 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 boats heading be more upstream, more downstream, or the same as in part (a)? Explain.

As two boats approach the marina, the velocity of boat 1 relative to boat 2 is 2.15 m/s in a direction 47.0east 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?

What speed must the dragony have if the line of sight, which is parallel to the xaxis 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

Suppose the dragony now approaches its prey along a path with , but it still keeps the line of sight parallel to the x axis. Is the speed of the dragony in this new case greater than, less than, or equal to its speed in Problem 79?

What is the correct motion camouage speed of approach for a dragony pursuing its prey at the angle  68.5? A. 0.295 m/s B. 0.698 m/s C. 0.806 m/s D. 2.05 m/s

If the dragony approaches its prey with a speed of 0.950 m/s, what angle is required to maintain a constant line of sight parallel to the x axis? A. 37.9 B. 38.3 C. 51.7 D. 52.1

Referring to Example 32 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.

Referring to Example 32 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 and 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?