1. Lecture 4 Perform basic vector algebra (addition and subtraction)
Today: Chapter 3
Introduce scalars and vectors
Perform basic vector algebra
(addition and subtraction)
Interconvert between
Cartesian & Polar coordinates 1

2. An “interesting” 1D motion problem: A raceq
Finish
Start
Two cars start out at a speed of 9.8 m/s. One travels along the horizontal at constant velocity while the second follows a 9.8 m long incline angled q below the horizontal down and then an identical incline up to the finish point. The acceleration of the car on the incline is g sin q (because of gravity & little g)
At what angles q, if any, is the race a tie?

3. A race According to our dynamical equations x(t) = xi + vi t + ½ a t2
v(t) = vi + a t
A trivial solution is q = 0° and the race requires 2 seconds.
Horizontal travel
Constant velocity so t = d / v = 2 x 9.8 m (cos q) / 9.8 m/s
t1 = 2 cos q seconds
Incline travel
9.8 m = 9.8 m/s ( t2 /2) + ½ g sin q (t2 /2)2
1 = ½ t2+ ½ sin q (½ t2)2  sin q (t2)2 + 4 t2 - 8 =0

4. Solving analytically is a challenge
A sixth order polynominal, (if z=1, =0)

5. Solving graphically is easier
Solve graphically
q=0.63 rad
q= 36°
If q < 36° then the incline is faster
If q < 36 ° then the horizontal track is faster

6. A science project
You drop a bus off the Willis Tower (442 m above the side walk). It so happens that Superman flies by at the same instant you release the bus. Superman is flying down at 35 m/s.
How fast is the bus going when it catches up to Superman?

7. A “science” project
You drop a bus off the Willis Tower (442 m above the side walk). It so happens that Superman flies by at the same instant you release the car. Superman is flying down at 35 m/s.
How fast is the bus going when it catches up to Superman?
Draw a picture y t yi

8. A “science” project Draw a picture Curves intersect at two points yyi
Draw a picture
Curves intersect at
two points

9. Coordinate Systems and Vectors
In 1 dimension, only 1 kind of system,
Linear Coordinates (x) +/-
In 2 dimensions there are two commonly used systems,
Cartesian Coordinates (x,y)
Circular Coordinates (r,q)
In 3 dimensions there are three commonly used systems,
Cartesian Coordinates (x,y,z)
Cylindrical Coordinates (r,q,z)
Spherical Coordinates (r,q,f)

10. Scalars and Vectors
A scalar is an ordinary number.
Has magnitude ( + or - ), but no direction
May have units (e.g. kg) but can be just a number
Represented by an ordinary character
Examples: mass (m, M) kilograms
distance (d,s) meters
spring constant (k) Newtons/meter

11. A = C A ≠B, B ≠ C Vectors act like…
Vectors have both magnitude and a direction
Vectors: position, displacement, velocity, acceleration
Magnitude of a vector A ≡ |A|
For vector addition or subtraction we can shift vector position at will (NO ROTATION)
Two vectors are equal if their directions, magnitudes & units match. A B C
A = C
A ≠B, B ≠ C

12. A or A A Vectors look like...
There are two common ways of indicating that something is a vector quantity:
Boldface notation: A
“Arrow” notation:
A or A A

13. A scalar can’t be added to a vector, even if they have the same units.
Scalars and Vectors
A scalar can’t be added to a vector, even if they have the same units.
The product of a vector and a scalar is another vector in the same “direction” but with modified magnitude B
A = B A

14. Exercise Vectors and Scalars
Which of the following is cannot be a vector ?
While I conduct my daily run, several quantities describe my condition
my velocity (3 m/s)
my acceleration downhill (30 m/s2)
my destination (the lab - 100,000 m east)
my mass (150 kg)

15. Vectors and 2D vector addition
The sum of two vectors is another vector.
A = B + C B B A C C

16. 2D Vector subtraction
Vector subtraction can be defined in terms of addition.
B - C
= B + (-1)C B B -C
B - C -C
B+C
Different direction and magnitude !

17. Unit Vectors û A Unit Vector points : a length 1 and no units
Gives a direction.
Unit vector u points in the direction of U
Often denoted with a “hat”: u = û
U = |U| û û
Useful examples are the cartesian unit vectors [ i, j, k ] or
Point in the direction of the x, y and z axes.
R = rx i + ry j + rz k or
R = x i + y j + z k y j x i k z

18. Vector addition using components:
Consider, in 2D, C = A + B.
(a) C = (Ax i + Ay j ) + (Bx i + By j ) = (Ax + Bx )i + (Ay + By )
(b) C = (Cx i + Cy j )
Comparing components of (a) and (b):
Cx = Ax + Bx
Cy = Ay + By
|C| =[ (Cx)2+ (Cy)2 ]1/2 C Bx A By B Ax Ay

19. Example Vector Addition
Vector B = {3,0,2}
Vector C = {1,-4,2}
What is the resultant vector, D, from adding A+B+C?
{3,-4,2}
{4,-2,5}
{5,-2,4}
None of the above

20. Converting Coordinate Systems (Decomposing vectors)
In polar coordinates the vector R = (r,q)
In Cartesian the vector R = (rx,ry) = (x,y)
We can convert between the two as follows: y
(x,y) r ry
tan-1 ( y / x )  rx x
In 3D

21. Motion in 2 or 3 dimensions
Position
Displacement
Velocity (avg.)
Acceleration (avg.)

22. Kinematics All this complexity is hidden away in
In 2-dim. position, velocity, and acceleration of a particle:
r = x i + y j
v = vx i + vy j (i , j unit vectors )
a = ax i + ay j
All this complexity is hidden away in
r = r(Dt) v = dr / dt a = d2r / dt2

23. Kinematics All this complexity is hidden away in
The position, velocity, and acceleration of a particle in
3-dimensions can be expressed as:
r = x i + y j + z k
v = vx i + vy j + vz k (i , j , k unit vectors )
a = ax i + ay j + az k
All this complexity is hidden away in
r = r(Dt) v = dr / dt a = d2r / dt2

24. Assignment: Read Chapter 4.1 to 4.3
Lecture 4
Assignment: Read Chapter 4.1 to 4.3 1