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The basic quantities you use to describe motion in two dimensions – displacement, velocity, and acceleration – are vectors. A vector is an object that has both a magnitude and a direction. When you have an equation that relates two vectors, you can break each vector into parts, called components. You end up with two equations, which are usually much easier to solve.

The Problems You’ll Work On

In this chapter on two-dimensional vectors and two-dimensional motion, you work with the following situations:

✓ Adding and subtracting vectors

✓ Multiplying a vector by a scalar

✓ Taking apart a vector to find its components

✓ Determining the magnitude and direction of a vector from its components

✓ Finding displacement, velocity, and acceleration in two dimensions

✓ Calculating the range and time of flight of projectiles

What to Watch Out For

While you zig and zag your way through the problems in this chapter, avoid running into obstacles by:

✓ Identifying the correct quadrant when finding the direction of a vector

✓ Finding the components before trying to add or subtract two vectors

✓ Breaking the displacement, velocity, and acceleration vectors into components to turn one difficult problem into two simple problems

✓ Remembering that the vertical component of velocity is zero at the apex

✓ Recognizing that the horizontal component of acceleration is zero for freely falling objects

Getting to Know Vectors

71–72

71. How many numbers are required to specify a two-dimensional vector?

72. Marcus drives 45 kilometers at a bearing of 11 degrees north of west. Which of the underlined words or phrases represents the magnitude of a vector?

Adding and Subtracting Vectors

73–75

73. Vector U points west, and vector V points north. In which direction does the resultant vector point?

74. If vectors A, B, and C all point to the right, and their lengths are 3 centimeters, 5 centimeters, and 2 centimeters, how many centimeters long is the resultant vector formed by adding the three vectors together?

75. Initially facing a flagpole, Jake turns to his left and walks 12 meters forward. He then turns completely around and walks 14 meters in the opposite direction. How many meters farther away from the flagpole would Jake have ended had he started his journey by turning to the right and walking 14 meters and then turning completely around and walking the final 12 meters?

Adding Vectors and Subtracting Vectors on the Grid

76–79

76. If and , what is the value of ?

77. If , what is the value of ?

78. Given that and , calculate .

79. Given the three vectors , , and , solve for D if .

Breaking Vectors into Components

80–83

80. Vector A has a magnitude of 28 centimeters and points at an angle 80 degrees relative to the x-axis. What is the value of ? Round your answer to the nearest tenth of a centimeter.

81. Vector C has a length of 8 meters and points 40 degrees below the x-axis. What is the vertical component of C, rounded to the nearest tenth of a meter?

82. Jeffrey drags a box 15 meters across the floor by pulling it with a rope. He exerts a force of 150 newtons at an angle of 35 degrees above the horizontal. If work is the product of the distance traveled times the component of the force in the direction of motion, how much work does Jeffrey do on the box? Round to the nearest ten newton-meters.

83. Three forces pull on a chair with magnitudes of 100, 60, and 140, at angles of 20 degrees, 80 degrees, and 150 degrees to the positive x-axis, respectively. What is the component form of the resultant force on the chair? Round your answer to the nearest whole number.

Reassembling a Vector from Its Components

84–87

84. What are the magnitude and direction of the vector ? Round your answers to the nearest tenth place and give your angle (direction) in units of degrees.

85. Given vectors and , what angle would vector C make with the x-axis if ? Round your answer to the nearest tenth of a degree.

86. If you walk 12 paces north, 11 paces east, 6 paces south, and 20 paces west, what is the magnitude (in paces) and direction (in degrees relative to the positive x-axis) of the resultant vector formed from the four individual vectors? Round your results to the nearest integer.

87. After a lengthy car ride from a deserted airfield to Seneca Airport, Candace finds herself 250 kilometers north and 100 kilometers west of the airfield. At Seneca, Candace boards a small aircraft that flies an unknown distance in a southwesterly direction and lands at Westsmith Airport. The next day, Candace flies directly from Westsmith to the airfield from which she started her journey. If the flight from Westsmith was 300 kilometers in distance and flew in a direction 15 degrees south of east, how many kilometers was Candace’s flight from Seneca to Westsmith? Round your answer to the nearest integer.

Describing Displacement, Velocity, and Acceleration in Two Dimensions

88–96

88. Hans drives 70 degrees north of east at a speed of 50 meters per second. How fast is Hans traveling northward? Round your answer to the nearest integer.

89. If you walk 25 meters in a direction 30 degrees north of west and then 15 meters in a direction 30 degrees north of east, how many meters did you walk in the north-south direction?

90. Jake wants to reach a postal bin at the opposite corner of a rectangular parking lot. It’s located 34 meters away in a direction 70 degrees north of east. Unfortunately, the lot’s concrete was recently resurfaced and is still wet, meaning that Jake has to walk around the lot’s edges to reach the bin. How many meters does he have to walk? Round your answer to the nearest whole meter.

91. If , use the following information to determine the components of D. Use ordered-pair notation rounded to the nearest tenth of a meter for your answer. (All angles are measured relative to the x-axis.)

A: 45 meters at 20 degrees

B: 18 meters at 65 degrees

C: 32 meters at –20 degrees

92. To walk from the corner of Broadway and Park Place to the corner of Church and Barkley in Central City, a person must walk 150 meters west and then 50 meters south. How many meters shorter would a direct route be? Round your answer to the nearest meter.

93. If Jimmy walks 5 meters east and then 5 meters south, what angle does the resultant displacement vector make with the positive x-axis (assuming the positive x-axis points east)?

94. A swimmer can move at a speed of 2 meters per second in still water. If he attempts to swim straight across a 500-meter-wide river with a current of 8 meters per second parallel to the riverbank, how many meters will the swimmer traverse by the time he reaches the other side? Round your answer to the nearest tenth of a kilometer.

95. A basketball rolling at a rate of 10 meters per second encounters a gravel patch that is 5 meters wide. If the basketball is moving in a direction 15 degrees north of east, and if the sides of the gravel patch are aligned parallel to the north-south axis, will the basketball still be rolling by the time it reaches the far side of the patch if the gravel gives it an acceleration of –3 meters per second squared in the same direction it started when it entered the patch? If so, what will be its speed in the easterly direction upon exiting the patch? If not, how many meters (measured perpendicularly to the western edge of the patch) will the basketball roll on the gravel before stopping? Round your numerical response to the nearest integer.

96. Partway through a car trip, a dashboard compass stops working. At the time it broke, Bill had driven 180 kilometers in a direction 70 degrees north of west. He then proceeds to drive 45 kilometers due south on the highway, before turning right and driving 18 kilometers west on Sunset St. When he stops the car, how far is Bill from the location where he began the trip? Round your answer to the nearest kilometer.

Moving under the Influence of Gravity: Projectile Launched Horizontally

97–99

97. A marble rolls off a 2-meter-high, flat tabletop. In how many seconds will it hit the floor? Round your answer to the nearest tenth of a second.

98. Mark rolls a boulder off a cliff located 22 meters above the beach. If he’s able to impart a velocity of 0.65 meters per second to the boulder, how many meters from the base of the cliff will the boulder land? Round your answer to the nearest tenth of a meter.

99. A car flies off a flat embankment with a velocity of 132 kilometers per hour parallel to the ground 45 meters below. With what velocity does the car ultimately crash into the ground? Round your answer to the nearest meters per second.

Moving under the Influence of Gravity: Projectile Launched at an Angle

100–105

100. Alicia kicks a soccer ball with a velocity of 10 meters per second at a 60-degree angle relative to the ground. What is the horizontal component of the velocity? Round your answer to the nearest tenth of a meter per second.

101. The punter for the San Diego Chargers kicks a football with an initial velocity of 18 meters per second at a 75-degree angle to the horizontal. What is the vertical component of the ball’s velocity at the zenith (highest point) of its path? Round your answer to the nearest tenth of a meter per second.

102. A cannonball fired at a 20-degree angle to the horizontal travels with a speed of 25 meters per second. How many meters away does the cannonball land if it falls to the ground at the same height from which it launched? Round your answer to the nearest meter.

103. Launching from a 100-meter-high ski jump of unknown inclination at 40 meters per second, an Olympic athlete grabs 8.2 seconds of hang time before landing on the ground. How far away from the jump does she land?

104. A cannon tilted at an unknown angle fires a projectile 300 meters, landing 11 seconds after launch at a final height equal to its starting one. At what angle was the cannon fired?

105. Will a baseball struck by a bat, giving the ball an initial velocity of 35 meters per second at 40 degrees to the horizontal, result in a home run if it must clear a 1.8-meter-high fence 120 meters away? If it will, by how many centimeters will the ball clear? If not, how many centimeters short will it be? Assume that the ball is struck at a height of 0.8 meters, and round your answer to the nearest 10 centimeters.