1. Lecture 4 Interconvert between Cartesian and Polar coordinates
Today: Ch. 3 (all) & Ch. 4 (start)
Perform vector algebra (addition and subtraction)
Interconvert between Cartesian and Polar coordinates
Work with 2D motion
Deconstruct motion into x & y or parallel & perpendicular
Obtain velocities
Obtain accelerations
Deduce components parallel and
perpendicular to the trajectory path 1

2. 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)

3. 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

4. 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

5. A scalar is an ordinary number.
Scalars
A scalar is an ordinary number.
A magnitude ( + or - ), but no direction
May have units (e.g. kg) but can be just a number
No bold face and no arrow on top.
The product of a vector and a scalar is another vector in the same “direction” but with modified magnitude B
A = B A

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

7. 2D Vector subtraction
Vector subtraction can be defined in terms of addition.
B - C
= B + (-1)C B C

8. 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 !

9. 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

10. 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

11. 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

12. 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

13. Decomposing vectors into components A mass on a frictionless inclined plane
A block of mass m slides down a frictionless ramp that makes angle  with respect to horizontal. What is its acceleration a ? m a 

14. Decomposing vectors into components A mass on a frictionless inclined plane
A block of mass m slides down a frictionless ramp that makes angle  with respect to horizontal. What is its acceleration a ? m
g sin  
g = - g j 
-g cos 

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

16. Instantaneous Velocity
But how we think about requires knowledge of the path.
The direction of the instantaneous velocity is along a line that is tangent to the path of the particle’s direction of motion. v

17. Average Acceleration
The average acceleration of particle motion reflects changes in the instantaneous velocity vector (divided by the time interval during which that change occurs).
Instantaneous
acceleration

18. Tangential ║ & radial acceleration= +
Acceleration outcomes:
If parallel changes in the magnitude of
(speeding up/ slowing down)
Perpendicular changes in the direction of
(turn left or right)

19. 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

20. x vs y x vs t y vs t Special Case x y y t x t
Throwing an object with x along the horizontal and y along the vertical.
x and y motion both coexist and t is common to both
Let g act in the –y direction, v0x= v0 and v0y= 0
x vs y t x 4
x vs t
y vs t
t = 0 y 4 y 4 t x

21. x vs y Another trajectory y
Can you identify the dynamics in this picture?
How many distinct regimes are there?
Are vx or vy = 0 ? Is vx >,< or = vy ?
x vs y
t = 0 y
t =10 x

22. x vs y Another trajectory y x
Can you identify the dynamics in this picture?
How many distinct regimes are there?
0 < t < < t < 7 7 < t < 10
I. vx = constant = v0 ; vy = 0
II. vx = vy = v0
III. vx = 0 ; vy = constant < v0
x vs y
t = 0
What can you say about the acceleration? y
t =10 x

23. Exercises 1 & 2 Trajectories with acceleration
A rocket is drifting sideways (from left to right) in deep space, with its engine off, from A to B. It is not near any stars or planets or other outside forces.
Its “constant thrust” engine (i.e., acceleration is constant) is fired at point B and left on for 2 seconds in which time the rocket travels from point B to some point C
Sketch the shape of the path
from B to C.
At point C the engine is turned off.
after point C (Note: a = 0)

24. Exercise 1 Trajectories with accelerationB C B C
From B to C ? A B A B C D
None of these B C B C C D

25. Exercise 2 Trajectories with acceleration
After C ? A B A B C D
None of these C C C D

26. Exercise 2 Trajectories with acceleration
After C ? A B A B C D
None of these C C C D

27. Assignment: Read all of Chapter 4, Ch. 5.1-5.3
Lecture 4
Assignment: Read all of Chapter 4, Ch 1