Chapter 3

Two-Dimensional Motion and Vectors

Section 3.1

Introduction to Vectors

Scalars and Vectors

• Scalar: a physical quantity that has only a magnitude but no direction

• Vector: a physical quantity that has both a magnitude and direction

Scalar or vector

• Are the following quantities scalars or vectors?– Weight– Displacement– Speed– Time– Velocity– Force

Scalars and Vectors

• In this book, vectors are indicated by the use of boldface type

• In this book, scalars are indicated by the use of italics

Vectors

• Vectors can be added graphically– They must describe similar quantities– They must have the same units

• The answer found by adding two vectors is called the resultant (vector sum)

Triangle method of addition

• Use a reasonable scale• Draw the tail of one vector starting at the tip of the other• Draw the resultant vector starting at the tail of the first

vector to the tip of the last vector

Vectors continued

• Vectors can be added in any order• Commutative property of vectors

Vectors continued

• To subtract a vector, add the opposite• When you multiply or divide a vector by a scalar, the

result is a vector– i.e. Twice as fast

• Always REMEMBER to indicate direction– North of east– South of west

Section 3.2

Vector Operations

Coordinate Systems in 2 Dimensions

• In Chapter 2 we looked at motion in 1 dimension

Coordinate Systems in Two Dimensions

• Adding another axis helps us describe two dimensional motion

• It also simplifies analysis of motion in one dimension– Fig 3.6

• In a, if the direction of the plane changes, we must turn the axis again

• It would also be difficult to describe another moving object if it is not traveling in a different direction

• Having 2 axes simplifies this significantly

Coordinate Systems in Two Dimensions

• There are no firm rules for applying coordinate systems• However, you must be consistent• It is usually best to use the system that makes solving

the problem the easiest• This is why we use the coordinate system

Determining Resultant Magnitude and Direction

• In Section 3.1 we did this graphically• This takes too long and is only accurate if you draw

VERY CAREFULLY• A better way is by using the Pythagorean theorem

Determining Resultant Magnitude and Direction

• In order to find the direction of the resultant, we can use the tangent function

Problem

• A Plane travels from Houston, Texas, to Washington, D.C., which is 1540 km east and 1160 km north of Houston. What is the total displacement of the plane?

Resolving Vectors into Components

• The horizontal and vertical parts that add up to give the resultant are called components.– The x component is parallel to the x-axis and the y component is

parallel to the y-axis

• Any vector can be described by a set of perpendicular components.– Fig 3.10

Resolving Vectors into Components

• For a refresher on trigonometry, see Appendix A in the back of the book.

sinθ =opphyp

cosθ =adjhyp

Problems

• An arrow is shot from a bow at an angle of 25o above the horizontal with an initial speed of 45 m/s. Find the horizontal and vertical components of the arrow’s initial velocity.

• The arrow strikes the target with a speed of 45 m/s at an angle of -25o with respect to the horizontal. Calculate the horizontal and vertical components of the arrow’s final velocity.

Adding vectors that are not perpendicular

• A plane travels 50km at an angle of 35o to the ground, then climbs at only 10o to the ground for 220 km.

• These vectors do not form a right triangle, so we can’t use the tangent function or the Pythagorean theorem when adding them

• However, we can resolve each of the plane’s displacement vectors into their x and y components

Adding vectors that are not perpendicular

• We can then use the Pythagorean theorem and tangent function on the vector sum of the two perpendicular components of the resultant

Adding Vectors Algebraically

• A hiker walks 25.5 km from her base camp at 35o south of east. On the second day, she walks 41.0 km in a direction 65o north of east, at which point she discovers a forest ranger’s tower. Determine the magnitude and direction of her resultant displacement between the base camp and the ranger’s tower.

Adding vectors algebraically

• Step 1: Select a coordinate system, draw a sketch of the vectors to be added and label each vector

Adding vectors Algebraically

• Step 3: Find the x and y components of the total displacement.

• Step 4: Use the Pythagorean theorem to find the magnitude of the resultant vector

• Step 5: Use a suitable trigonometric function to find the angle the resultant vector makes with the x-axis

• Step 6: Evaluate your answer.

HW Assignment

• Pg 91: Practice 3a - 1, 2• Pg 94: Practice 3b - 2, 4, 6• Pg 97: Practice 3c - 2, 3, 4

Section 3.3

Projectile Motion

Two-Dimensional Motion

• The velocity, acceleration and displacement of an object thrown into the air don’t all point in the same direction.

• We resolve vectors into components to make it simpler.• At the end, we can recombine the components to

determine the resultant.

Components simplify motion

• As a long jumper approaches his jump, he runs in a straight line.

• When he jumps, his velocity has both horizontal and vertical components.

• To analyze his motion we apply the kinematic equations to one direction at a time

Projectile Motion

• Free fall with an initial horizontal velocity• Projectiles follow parabolic trajectories • However, there is air resistance at the surface of the

earth and horizontal velocity slows down.• In this class, we will consider the horizontal velocity to be

constant.

Vertical motion of a projectile

Vertical motion of a projectile that falls from rest

vy , f =−gΔt

vy , f2 =−2gΔy

Δy =1

2g(Δt)

2

Horizontal motion of a projectile

Δx = vxΔt

vx =vx, i =constant

Problem

• People in movies often jump from buildings into pools. If a person jumps from the 10th floor (30.0 m) to a pool that is 5.0 m away from the building, with what initial velocity must the person jump?

Objects launched at an angle

• For an object launched at an angle, the sine and cosine functions can be used to find the horizontal and vertical components of the vi.

• Use components to analyze objects launched at an angle

vx =vi(cosθ) =constantΔx = vi(cosθ )Δtvy , f =vi(sinθ) −gΔt

vy , f2 =vi2(sinθ)2 −2gΔy

Δy = vi(sinθ)Δt −1

2g(Δt)

2

Problem

• A zookeeper finds an escaped monkey hanging from a light pole. Aiming her tranquilizer gun at the monkey, the zookeeper kneels 10.0 m from the light pole, which is 5.00 m high. The tip of her gun is 1.00 m above the ground. The monkey tries to trick the zookeeper by dropping a banana, then continues to hold onto the light pole. At the moment the monkey releases the banana, the zookeeper shoots. If the tranquilizer dart travels at 50.0 m/s, will the dart hit the monkey, the banana, or neither one?

Problem

• A golfer practices driving balls off a cliff and into water below. The cliff is 15 m from the water. If the golf ball is launched at 51 m/s at an angle of 15o, how far does the ball travel horizontally before hitting the water?