1. 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

2. 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”

3. 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

4. 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.

5. 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

6. 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”.

7. 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

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

9. 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

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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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.

17. 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.

18. 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

19. 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

20. 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 ° N of W
37.5 km
Next 1.5 Hours:
123 km
Travels 37.5 N
60°
Island

21. 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

22. 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

23. “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”

24. “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”

25. 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).

26. 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.

27. Reading Assignment
Next class: 3.4 – 3.5