Unit 3Two-Dimensional Motion and Vectors

3-1 Introduction to Vectors

•In Unit 1, motion was one-dimensional

• To show forward, right or upward motion, we used a positive sign

• To show backward, left or downward motion, we used a negative sign

Scalars and Vectors

• Scalar- A quantity that can be completely specified by its magnitude with appropriate units; in other words, it has a magnitude but no direction

• Ex: speed, volume, area, mass

• Vector- A physical quantity that has a magnitude and direction

• Ex: displacement, velocity, acceleration

• Direction is based on compass

• Ex: North, South, Northeast, Southwest• In textbooks, vectors are represented by

boldface print; scalars are represented by italics

Vectors in Diagrams

•Vectors are shown as arrows that point in the direction of the vector; length of arrow is related to the vector’s magnitude

Adding Vectors Graphically

•Before adding vectors, make sure they measure the same quantities

•Ex: both measure velocity

•Before adding vectors, make sure they both have the same units

•Ex: both measured in meters (as opposed to one in meters and the other in kilometers)

Adding Vectors

• Resultant- A vector representing the sum of two or more vectors

Ex: Vectors can be added to find total displacement

Steps to Adding Vectors Graphically(Triangle Method of Addition)

1) Draw situation to scale on paper

2) Draw first vector with correct direction and magnitude

3) Draw second vector with tail at the head end of the first arrow; continue until all vectors are drawn

4) Resultant vector goes from tail of first arrow to head of last arrow

5) Multiply resultant length by scale to determine magnitude; use a protractor to determine direction

Vector Addition Examples

Properties of Vectors

1) Vectors can be moved parallel to themselves in a direction; vectors can be drawn at any place in a diagram so long as they are parallel to their starting position and equal in length.

• Since a and b are parallel and

• even in length, they are equal

• in magnitude and direction.

2) Vectors can be added in any order; resultant vector will always be the same

• Consider 2 vectors:

• Add B to A or A to B– resultant is the same!

3) To subtract a vector, add its opposite

- A negative vector is the same in magnitude but opposite in direction.

- Ex: Negative vector of the velocity of a car moving 30 m/s to the right (+30 m/s) is 30 m/s to the left (-30 m/s)

4) Multiplying or dividing vectors by scalars results in vectors

3-2 Vector Operations• Coordinate systems in two dimensions- Movement of an

object can be divided into a horizontal component (x) and a vertical component (y)

Vector Components

Determining Resultant Magnitude and Direction

• Use Pythagorean Theorem to find the magnitude of the resultant

• (Length of hypotenuse)2 = (Length of one leg)2 +

• (Length of other leg)2

• Component- Horizontal and vertical parts that add up to give the actual displacement of an object

Resultant Direction in Right Triangles

• SOH- CAH- TOA:

• Sine θ = opp

• hyp

• Cosine θ = adj

• hyp

• Tangent θ = opp

• adj

3-3 Projectile Motion

• Components can be used to determine the resultant; simpler than vector multiplication

• Components simplify projectile motion because it allows us to analyze one direction at a time

• Projectile Motion- free-fall with an initial horizontal velocity

• Objects thrown or launched into the air are called projectiles (including people!)

• Projectiles move in a parabolic trajectory (curve)

• Horizontal motion is independent of vertical motion

• Free Fall versus Projectile

• **Note: graph of a projectile looks almost like the graph of an object in free fall, but curve is more spread out

Fast-Moving Projectiles—Satellites• What if a ball were thrown so fast that the curvature of Earth

came into play?

• If the ball was thrown fast enough to exactly match the curvature of Earth, it would go into orbit

• Satellite – a projectile moving fast enough to fall around Earth rather than into it (v = 8 km/s, or 18,000 mi/h)

• Due to air resistance, we launch our satellites into higher orbits so they will not burn up

Satellites

Launch Speed less than 8000 m/s Projectile falls to Earth

                                             

Launch Speed less than 8000 m/s Projectile falls to Earth

                                             

Launch Speed equal to 8000 m/s Projectile orbits Earth - Circular Path

                                             

Launch Speed greater than 8000 m/s Projectile orbits Earth - Elliptical Path

                                           

Free Fall versus Projectile

Projectile Motion

Projectile Motion

No Gravity With Gravity

Projectile Motion

Projectile Motion Graph

Projectile Motion Formulas

• Vertical Motion of a Projectile that Falls from rest:

• Vy,f = -gΔt

• Vy,f2 = -2gΔy

• Δy = -1/2 g (Δt)2

• Horizontal Motion of a Projectile:

• vx = vx, o = constant

• Δx = vxΔt

3-4 Relative Motion

•Frame of Reference- a coordinate system for specifying the precise location of objects in space

•Velocity, displacement, and acceleration depend on the frame of reference

The Plane and the PackageHow would an object appear to be moving if it were dropped from a moving plane?