Two-Dimensional Motion and Vectors Section 1

Preview

Section 1 Introduction to Vectors

Section 2 Vector Operations

Section 3 Projectile Motion

Section 4 Relative Motion

Two-Dimensional Motion and Vectors Section 2

Vector Operations

• Use a traditional x-y coordinate system as shown below on the right.• The Pythagorean theorem and tangent function can be used to add vectors.

– More accurate and less time-consuming than the graphical method

Two-Dimensional Motion and Vectors Section 2

Pythagorean Theorem and Tangent Function

Two-Dimensional Motion and Vectors Section 2

Vector Addition - Sample Problems

• 12 km east + 9 km east = ?– Resultant: 21 km east

• 12 km east + 9 km west = ?– Resultant: 3 km east

• 12 km east + 9 km south = ?– Resultant: 15 km at 37° south of east

• 12 km east + 8 km north = ?– Resultant: 14 km at 34° north of east

Two-Dimensional Motion and Vectors Section 2

Resolving Vectors Into Components

Two-Dimensional Motion and Vectors Section 2

Resolving Vectors into Components• Opposite of vector addition• Vectors are resolved into x and y components

• For the vector shown at right, find the vector components vx (velocity in the x direction) and vy (velocity in the y direction). Assume that that the angle is 20.0˚.

• Answers:– vx = 89 km/h

– vy = 32 km/h

Two-Dimensional Motion and Vectors Section 2

Adding Non-Perpendicular Vectors

• Four steps– Resolve each vector into x and y components– Add the x components (xtotal = x1 + x2)– Add the y components (ytotal = y1 + y2)– Combine the x and y totals as perpendicular vectors

Two-Dimensional Motion and Vectors Section 2

Click below to watch the Visual Concept.

Visual Concept

Adding Vectors Algebraically

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Displacement

• Following the directions of the path, a pirate walks 45.0 m north, then turns and walks 7.5 m east. What single straight line displacement could the pirate have taken to reach the destination.

• Answer– 4.5m at 9.5° E of N

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Displacement

• Emily passes a soccer ball 6.0 m directly across the field to Kara, who then kicks the ball 14.5m directly down the field to Luisa. What is the ball’s total displacement as it travels between Emily and Luisa?

• Answer– 15.7m at 22° to the side of down the field

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Displacement

• A humming bird flies 1.2 m along a straight path at a height of 3.4 m above the ground. Upon spotting a flower below, the humming bird drops directly downward 1.4 m to hover in front of the flower. What is the hummingbird’s displacement?

• Answer– 1.8 m at 49° below the horizontal

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Resolving Vectors

• Find the horizontal and vertical component of the 125 m displacement of a superhero who flies down from the top of a tall building at an angle of 25 ° below the horizontal.

Answer– 110 m– -53 m

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Resolving Vectors

• Find the horizontal and vertical component of the child’s toboggan ride down a hill if the angle of the hill is 30.5 ° to the horizontal and the hill is 23m long.

Answer– 19.8 m– -11.7 m

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Resolving Vectors

• Find the horizontal and vertical component of the truck’s velocity, if the truck drives up a hill with a 15° incline at a constant speed of 22 m/s.

Answer– 21m/s– 5.7m/s

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Algebraically

• A football player runs directly down the field for 35 m before turning to the right at an angle of 25° from his original direction an running an additional 15 m before getting tackled. What is the total displacement of the player

Answer– 49 m at 7.3° to the right of down field

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Algebraically

• A plane travels 25 km at an angle of 35 to the ground, then changes direction and travels 515 km at an angle of 22° to the ground. What is the displacement of the plane?

Answer– 540,000 m at 22° above the horizontal

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Algebraically

• During a rodeo a clown runs 8.0 m north, turns 35 ° east of north, and runs 3.5 m. then , after waiting for the bull to come near, the clown turns due east and runs 5.0 m to exit the arena. What is the clown’s displacement?

Answer– 13.5 m at 37° north of east

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Resolving Vectors

• Find the magnitude resultant velocity.

A fish swimming a t 3.0 m/s across a river that moves at 5.0 m/s?

What is the direction?• Answer

– 5.8 m/s– 59° downriver from its intended path

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Resolving Vectors

• Find the magnitude resultant velocity.

A surfer travelling at 1.0 m/s across a wave that is traveling at 6.0 m/s?

What is the direction?• Answer

– 6.1 m/s– 9.5° from the direction the wave is traveling

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Resolving Vectors

• Find the component vectors along the directions noted in parentheses.

A car displaced northeast by 10.0 km (north and east

• Answer– 7070 m, 7070 m

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Resolving Vectors

• Find the component vectors along the directions noted in parentheses.

A duck accelerating away from a hunter at 2.0 m/s2 at an angle of 35° to the ground (horizontal and vertical)

• Answer– 1.6 m/s2, 1.1 m/s2

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Resolving Vectors

• Find the component vectors along the directions noted in parentheses.

A submarine moving at 10.0 m/s toward the surface at an angle of 35 ° to the ground (horizontal and vertical)

• Answer– 8.2 m/s, 5.7 m/s

Two-Dimensional Motion and Vectors Section 2

Classroom Practice Resolving Vectors

• Find the resultant displacement of a fox searching for mice in a prairie. First the fox heads 55° north of west for 10.0 m then it turns and heads west for 5.0m.

• Answer– 13.5 m at 37° north of east