Sect. 3-4: Analytic Method of Addition

Resolution of vectors into components:

YOU MUST KNOW &

UNDERSTAND

TRIGONOMETERY TO

UNDERSTAND THIS!!!!

Vector Components • Any vector can be expressed as the sum of two other vectors, called its

components. Usually, the other vectors are chosen so that they are perpendicular to each other.

• Consider vector V in a plane (say the xy plane)

• V can be expressed in terms of the Components Vx , Vy

• Finding the Components Vx & Vy is Equivalent to finding 2 mutually

perpendicular vectors which, when added (with vector addition) will give V.

• That is, find Vx & Vy such that

V Vx + Vy

(Vx || x axis, Vy || y axis)

Finding Components “Resolving into components”

• Mathematically, a component is a projection of a vector along an axis– Any vector can be

completely described by its components

• It is useful to use rectangular components– These are the projections of the vector along the x- and y-axes

When V is resolved into components: Vx & Vy

V Vx + Vy (Vx || x axis, Vy || y axis)

By the parallelogram method, the vector sum is: V = V1 + V2

In 3 dimensions, we also need a component Vz.

Brief Trig Review • Adding vectors in 2 & 3 dimensions using components requires

TRIGONOMETRY FUNCTIONS• HOPEFULLY, A REVIEW!!

– See also Appendix A!!• Given any angle θ, we can construct a right triangle:

Hypotenuse h, Adjacent side a, Opposite side o

ho

a

• Define trig functions in terms of h, a, o:

= (opposite side)/(hypotenuse)

= (adjacent side)/(hypotenuse) = (opposite side)/(adjacent side)

[Pythagorean theorem]

Signs of sine, cosine, tangent • Trig identity: tan(θ) = sin(θ)/cos(θ)

Trig Functions to Find Vector Components

[Pythagorean theorem]

We can & will use all of this to add vectors analytically!

Example

V = displacement 500 m, 30º N of E

Using Components to Add Two Vectors• Consider 2 vectors, V1 & V2. We want V = V1 + V2

• Note: The components of each vector are really one-dimensional vectors, so they can be added arithmetically.

“Recipe” (for adding 2 vectors using trig & components)

1. Sketch a diagram to roughly add the vectors graphically. Choose x & y axes. 2. Resolve each vector into x & y components using sines & cosines.

That is, find V1x, V1y, V2x, V2y. (V1x = V1cos θ1, etc.)

4. Add the components in each direction. (Vx = V1x + V2x, etc.)

5. Find the length & direction of V, using:

We want the vector sum V = V1 + V2

Example 3-2 A rural mail carrier leaves the post office & drives 22.0 km in a northerly direction. She then drives in a direction 60.0° south of east for 47.0 km. What is her displacement from the post office?

Example 3-3 A plane trip involves 3 legs, with 2 stopovers: 1) Due east for 620 km, 2) Southeast (45°) for 440 km, 3) 53° south of west, for 550 km. Calculate the plane’s total displacement.

Sect. 3-5: Unit Vectors • Convenient to express vector V in terms of it’s components

Vx, Vy, Vz & UNIT VECTORS along x,y,z axes

UNIT VECTOR a dimensionless vector, length = 1

• Define unit vectors along x,y,z axes:

i along x; j along y; k along z

|i| = |j| = |k| = 1

• Vector V. Components Vx, Vy, Vz :

Simple Example • Position vector r in x-y plane.

Components x, y:

r x i + y j

Figure

Vector Addition Using Unit Vectors • Suppose we want to add two vectors V1 & V2 in x-y plane:

V = V1 + V2

“Recipe” 1. Find x & y components of V1 & V2 (using trig!)

V1 = V1xi + V1yj V2 = V2xi + V2yj

2. x component of V: Vx = V1x + V2x

y component of V: Vy = V1y + V2y

3. So

V = V1 + V2 = (V1x+ V2x)i + (V1y+ V2y)j

Example 3-4 Rural mail carrier again. Drives 22.0 km in North. Then 60.0° south of east for 47.0 km. Displacement?

Another Analytic Method • Laws of Sines & Law of Cosines from trig.• Appendix A-9, p A-4, arbitrary triangle:

• Law of Cosines: c2 = a2 + b2 - 2 a b cos(γ)• Law of Sines: sin(α)/a = sin(β)/b = sin(γ)/c

α

β

γ

a

b

c

• Add 2 vectors: C = A + B

• Law of Cosines: C2 = A2 + B2 -2 A B cos(γ)

Gives length of resultant C.

• Law of Sines: sin(α)/A = sin(γ)/C, or sin(α) = A sin(γ)/C

Gives angle α

B

A

γ

β

α

AC

B

Given A, B, γ

Sect. 3-6: Kinematics in 2 Dimensions

Sect. 3-6: Vector Kinematics

Motion in 2 & 3 dimensions. Interested in displacement, velocity, acceleration. All are vectors. For 2- & 3-dimensional kinematics, everything is the same as in 1-dimensional motion, but we must now use full vector notation.

Position of an object (particle) is

described by its position vector, r. Displacement of the object is the change in its position:

Particle Path

Displacement takes time Δt

Velocity: Average Velocity = ratio of displacement Δr time interval Δt for the displacement:

vavg

vavg is in the direction of Δr. vavg independent of the path taken because Δr is also. In the limit as Δt & Δr 0 vavg Instantaneous Velocity v.

Direction of v at any point in a particle’s path is along a line tangent to the path at that point & in motion direction. The magnitude of the instantaneous velocity vector v is the speed, which is a scalar quantity.

Acceleration: Average Acceleration = ratio of the change in the instantaneous velocity vector divided by time during which the change occurs.

aavg

aavg is in the direction of

In the limit as Δt & 0 aavg Instantaneous Velocity a.

Various changes in particle motion may cause acceleration: 1) Magnitude velocity vector may change. 2) Direction of velocity vector may change. even if magnitude remains constant. 3) Both may change simultaneously.

Using unit vectors:

Kinematic Equations for 2-Dimensional Motion

• In the special case when the 2-dimensional motion has a constant acceleration a, kinematic equations can be developed that describe the motion

• Can show that these equations (next page) are similar to those of 1-dimensional kinematics.

• Motion in 2 dimensions can be treated as 2 independent motions in each of the 2 perpendicular directions associated with the x & y axes– Motion in y direction does not affect motion in x direction &

motion in x direction does not affect the motion in y direction.

– Results are shown on next page.

Generalization of 1-dimensional equations for constant acceleration to 2 dimensions. These are valid for constant acceleration ONLY!