Lecture 4
Today:
Chapter 3
 Introduce scalars and vectors
 Perform basic vector algebra
(addition and subtraction)
 Interconvert between
Cartesian & Polar coordinates
Physics 201: Lecture 4, Pg 1
An “interesting” 1D motion problem: A race
Start
q
q
Finish
 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?
Physics 201: Lecture 4, Pg 2
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  16  32 sin q
- 2  4  8 sin q
t2 

2 sin q
sin q
Physics 201: Lecture 4, Pg 3
Solving analytically is a challenge
- 2  4  8 sin q
2 cos q 
sin q
2 cos q  sin q cos q - 2  0
2
Let
z  cos q and 1-z  sin q
2
Solve : z - z  4 z - 8 z  4  0
6
4
2
A sixth order polynominal, (if z=1, 1-1+4-8+4 =0)
Physics 201: Lecture 4, Pg 4
Solving graphically is easier
- 2  4  8 sin q
t1  2 cos q  t2 
sin
q
Solve graphically
2.0
-2
time (sec.)
1.6
1.2
q=0.63 rad
q= 36°
4  8 sin q
sin q
2 cos q
0.8
0.4
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Angle (radians)
1.4
If q < 36° then the incline is faster
If q < 36 ° then the horizontal track is faster
Physics 201: Lecture 4, Pg 5
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?

Physics 201: Lecture 4, Pg 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 car. Superman
is flying down at 35 m/s.
 How fast is the bus going when it catches up to
Superman?

yi

Draw a picture
y
0
t
Physics 201: Lecture 4, Pg 7
A “science” project
yi
Draw a picture
 Curves intersect at
two points

y
0
g 2
y  - t  vSuperman t
2
g
2
- t  vSuperman
t  - vSuperman
2
g
2
v bus  - gt  g vSuperman  2vSuperman
g
Physics 201: Lecture 4, Pg 8
t
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)
Physics 201: Lecture 4, Pg 12
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)
distance (d,s)
spring constant (k)
kilograms
meters
Newtons/meter
Physics 201: Lecture 4, Pg 13
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=C
A
B
C
A ≠B, B ≠ C
Physics 201: Lecture 4, Pg 14
Vectors look like...

There are two common ways of indicating that
something is a vector quantity:
Boldface notation: A
A or A
“Arrow” notation:
A
Physics 201: Lecture 4, Pg 15
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
A = -0.75 B
A
B
Physics 201: Lecture 4, Pg 16
Exercise
Vectors and Scalars
While I conduct my daily run, several quantities describe my
condition
Which of the following is cannot be a vector ?
A.
B.
C.
D.
my velocity (3 m/s)
my acceleration downhill (30 m/s2)
my destination (the lab - 100,000 m east)
my mass (150 kg)
Physics 201: Lecture 4, Pg 17
Vectors and 2D vector addition

The sum of two vectors is another vector.
B
C
A =B+C
B
A
C
Physics 201: Lecture 4, Pg 18
2D Vector subtraction

Vector subtraction can be defined in terms of addition.
B-C
= B + (-1)C
B
B-C
-C
-C
B
B+C
Different direction
and magnitude !
Physics 201: Lecture 4, Pg 19
Unit Vectors



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 [ xˆ , yˆ , zˆ ]
x, y and z axes.
R = rx i + ry j + rz k
k
or
z
R=xi+yj+zk
y
j
i
x
Physics 201: Lecture 4, Pg 20
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
B
A
Ay
By
Bx
Ax
Physics 201: Lecture 4, Pg 21
Example
Vector Addition



Vector A = {0,2,1}
Vector B = {3,0,2}
Vector C = {1,-4,2}
What is the resultant vector, D, from adding A+B+C?
A.
B.
C.
D.
{3,-4,2}
{4,-2,5}
{5,-2,4}
None of the above
Physics 201: Lecture 4, Pg 22
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:
rx  x  r cos q
y
ry  y  r sin q

r  x ˆi  y ˆj
r x y
2
(x,y)
ry
q
rx
2
q  tan-1 ( y / x )
• In 3D
r
r x y z
2
2
x
2
Physics 201: Lecture 4, Pg 23
Motion in 2 or 3 dimensions


 Position
ri , ti and rf , t f
  
 Displacement
r  rf - ri


r

 Velocity (avg.)
v avg. 
t


v

 Acceleration (avg.)
a avg. 
t
Physics 201: Lecture 4, Pg 24