Last Time: One-Dimensional Motion with Constant Acceleration Freely Falling Objects Wraps up our discussion of 1D motion … Today: Vector Techniques: Increasingly Important How to Manipulate Vectors Two-Dimensional Motion HW #2 due Thurs, Sept 9, 11:59 p.m. Recitation Quiz #2 tomorrow

Vectors vs. Scalars All of the physical quantities we will encounter in PHY 211 can be categorized as either a vector or a scalar. Vectors have both a magnitude (size) and a direction. Scalars have only a magnitude. Examples of Scalars Speed Temperature (e.g., −40°F) Mass Time Interval Volumes, Areas Ordinary Arithmetic Examples of Vectors Displacement Δx Velocity Acceleration Momentum Angular Momentum “Vector Arithmetic”

a Representing a Vector B = magnitude of B A = magnitude of A We denote a vector mathematically in an equation as: a Example: scalar We draw vectors as arrows Points in the direction of the vector Length indicates the magnitude. Magnitude is denoted with the variable name, without the arrow above it. B = magnitude of B A = magnitude of A

Vector Equality Two vectors A and B are equal if : They have the same magnitude; and They have the same direction. x y These four vectors are all equal. A vector can be translated (or moved) parallel to itself without being affected.

Vector Addition: Geometric Important: Make sure the vectors have the same units !! Geometrically: “Triangle Method of Addition” of A and B Draw A with its direction specified relative to some coordinate system. Draw B with the tail of B starting at the tip of A. The resultant vector R = A + B is the vector drawn from the tail of A to the tip of B. y x

Commutative Law of Addition Suppose have two vectors A and B. Does A + B = B + A ? y y x x Yes. Vector addition is “commutative”.

Negative of a Vector A –A The negative of a vector A is defined to be the vector that when added to A, yields a resultant vector of 0. Thus, A and –A have the same magnitude, but opposite directions. y A –A x

Vector Subtraction The operation A – B is defined to be y y x x

Multiplying and Dividing by a Scalar We can multiply or divide a vector by a scalar. (Recall, a scalar is just a number.) These processes yield a vector. c = scalar (number) c = scalar (number) Example: Example: A A Two times the magnitude, but in same direction !! Half of the magnitude, but in same direction !! B B

Note You cannot “multiply” or “divide” two vectors !

Addition of > 2 Vectors The same rules for vector addition apply in the addition of more than 2 vectors. Example: B B A A C C R

Example Vector A points 15 units in the +x direction. Vector B points 15 units in the +y direction. Find the magnitude and direction of: A – B

Example A car travels 50 km in the east direction, and then 100 km in the northeast direction. Using vectors, find the magnitude and direction of a single vector that gives the car’s displacement relative to its starting point. y, North 100 km 50 km 45° x, East

Conceptual Question (p. 75) If B is added to A, under what condition does the resultant vector have a magnitude equal to (A + B) ? i.e., the sum of the magnitudes of A and B

Vector Components A = Ax + Ay Ay Ax “Components”: Ax = A cosθ The standard method of adding vectors makes use of the projections of the vectors along the axes of a coordinate system. These projections are called the components. A vector can be completely specified by its components. “Components”: A = Ax + Ay Ax = A cosθ “component vectors” Ay = A sinθ Ay θ Ax cos θ and sin θ determine the signs of Ax and Ay

Example A velocity vector has a magnitude of 10 m/s and a direction of 135° counter-clockwise from the +x-axis. Calculate the x- and y-components of this velocity vector.

Example A vector has an x-component of 5 units, and a y-component of –7 units. Find the magnitude and direction of the vector.

Adding Vectors Algebraically Suppose: R = A + B . The components of the resultant vector are given by: x y A Ay Ax x B By y Bx y + = By + Ay x Ax + Bx

Adding Vectors Algebraically Suppose: R = A + B . The components of the resultant vector are given by: x y A Ay Ax x B By y Bx y + = By + Ay x Ax + Bx

Example: Problem 3.17 The eye of a hurricane passes over an island in a direction of 60.0° north of west with a speed of 41.0 km/hr. Three hours later, the hurricane suddenly shifts north, and its speed slows to 25 km/hr. How far from the island is the hurricane 4.50 hours after it passes over the island? First 3 Hours: Travels 123 km @ 60° N of W 37.5 km Next 1.5 Hours: 123 km Travels 37.5 km @ N 60° Island

Why Do We Need Vectors ? Why did we spend all this time discussing vectors ? In 1D-motion, vector quantities (velocity, acceleration) can be taken into account by specifying either a + or – sign. In 2D- (or higher-dimensional) motion, this simple interpretation no longer works, and we must use vectors. 1D Example : 2D Example : If you make a series of steps along the x-axis, displacement from the origin is just the sum of the individual steps (taking account of signs). If you make a series of steps in the x-y plane, can’t just add up the magnitudes to find displacement from the origin! y x x

Displacement in 2D In 2D, an object’s displacement is defined to be the change in its position vector : : position vector at time ti SI Unit: m : position vector at time tf y Object moving along this path in the x-y coordinate system Final position rf is just : final initial displacement x

“value of the quantity as Δt becomes very small” Velocity in 2D In 2D, an object’s average velocity during a time interval Δt is its displacement divided by Δt: SI Unit: m/s This is a vector, just like the displacement ! In 2D, an object’s instantaneous velocity is the limit of its average velocity as Δt becomes very small: SI Unit: m/s This is a vector, just like the displacement ! “value of the quantity as Δt becomes very small”

“value of the quantity as Δt becomes very small” Acceleration in 2D In 2D, an object’s average acceleration during a time interval Δt is the change in its velocity divided by Δt: SI Unit: m/s2 This is a vector, just like the velocity ! In 2D, an object’s instantaneous acceleration is the limit of its average acceleration as Δt becomes very small: SI Unit: m/s2 This is a vector, just like the displacement ! “value of the quantity as Δt becomes very small”

Comment: Acceleration in 2D The acceleration is a vector, just like the velocity vector. If the magnitude of the velocity stays the same (speed), the velocity vector will still change if the direction changes ! In 2D, can have a non-zero acceleration if speed stays the same, but direction changes (example: driving in a circle at constant speed).

Good News When dealing with vectors, we can usually break the vectors down into their x- and y-components. Motion in 2D can be then be thought of as two separate 1D problems, along the x- and y-axes.

Reading Assignment Next class: 3.4 – 3.5