Physics 201: Lecture 4, Pg 1 Lecture 4 Today: Chapter 3 Introduce scalars and vectors Perform basic vector algebra (addition and subtraction) Interconvert between Cartesian & Polar coordinates
Jan 03, 2016
Physics 201: Lecture 4, Pg 1
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 2
An “interesting” 1D motion problem: A race
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 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 (because of gravity & little g)
At what angles , if any, is the race a tie?
9.8 m 9.8 m
FinishStart
Physics 201: Lecture 4, Pg 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 = 0° and the race requires 2 seconds. Horizontal travel
Constant velocity so t = d / v = 2 x 9.8 m (cos ) / 9.8 m/s t1 = 2 cos seconds
Incline travel
9.8 m = 9.8 m/s ( t2 /2) + ½ g sin t2 /2)2
1 = ½ t2+ ½ sin (½ t2)2 sin t2)2 + 4 t2 8 =0
sinsin842
sin2sin32164
2
t
Physics 201: Lecture 4, Pg 4
Solving analytically is a challenge
02cossincos2 2
A sixth order polynominal, (if z=1, 1-1+4-8+4 =0)
sin1 and cosLet 2 -zz
sin
sin842cos2
0484 :Solve 246 zzzz
Physics 201: Lecture 4, Pg 5
Solving graphically is easier
sinsin842
cos2 21
ttSolve graphically
0.2 1.00.60.40.0 1.41.20.8Angle (radians)
time
(se
c.)
1.6
1.2
0.8
0.4
0
2.0
cos2
sin
sin842
=0.63 rad= 36°
If < 36° then the incline is fasterIf < 36 ° then the horizontal track is faster
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 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 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 picturey
t
yi
0
Physics 201: Lecture 4, Pg 8
A “science” project
Draw a picture Curves intersect at
two points
y
t
yi
0
ttg
y Superman2 v
2
Supermanv 2
tg
Supermanv2g
t
SupermanSupermanbus v2v2
v g
gtg
Physics 201: Lecture 4, Pg 12
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,)
In 3 dimensions there are three commonly used systems,Cartesian Coordinates (x,y,z)Cylindrical Coordinates (r,,z)Spherical Coordinates (r,)
Physics 201: Lecture 4, Pg 13
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) kilogramsdistance (d,s) metersspring constant (k)
Newtons/meter
Physics 201: Lecture 4, Pg 14
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
Physics 201: Lecture 4, Pg 15
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
Physics 201: Lecture 4, Pg 16
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 BA = -0.75 B
Physics 201: Lecture 4, Pg 17
Exercise Vectors and Scalars
A. my velocity (3 m/s)B. my acceleration downhill (30 m/s2)C. my destination (the lab - 100,000 m east)D. my mass (150 kg)
Which of the following is cannot be a vector ?
While I conduct my daily run, several quantities describe my condition
Physics 201: Lecture 4, Pg 18
Vectors and 2D vector addition
The sum of two vectors is another vector.
A = B + CB
C A
B
C
Physics 201: Lecture 4, Pg 19
2D Vector subtraction
Vector subtraction can be defined in terms of addition.
B - C
B
-C
B
-C
B - C
= B + (-1)C
B+C Different direction and magnitude !
Physics 201: Lecture 4, Pg 20
Unit Vectors
A Unit Vector points : a length 1 and no units Gives a direction. Unit vector uu points in the direction of UU
Often denoted with a “hat”: u = û U = |U| û
û
x
y
z
ii
jj
kk
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
]ˆ,ˆ,ˆ[ zyx
Physics 201: Lecture 4, Pg 21
Vector addition using components:
Consider, in 2D, C = A + B. (a) CC = (Ax ii + Ay jj ) + (Bx i i + By jj ) = (Ax + Bx )ii + (Ay + By )
(b) CC = (Cx ii + Cy jj )
Comparing components of (a) and (b):
Cx = Ax + Bx
Cy = Ay + By
|C| =[ (Cx)2+ (Cy)2 ]1/2
CC
BxAA
ByBB
Ax
Ay
Physics 201: Lecture 4, Pg 22
ExampleVector Addition
A.A. {{3,-4,2}
B. {4,-2,5}
C. {5,-2,4}
D. None of the above
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?
Physics 201: Lecture 4, Pg 23
Converting Coordinate Systems (Decomposing vectors)
In polar coordinates the vector R = (r,)
In Cartesian the vector R = (rx,ry) = (x,y)
We can convert between the two as follows:
• In 3D
tan-1 ( y / x )
22 yxr
y
x
(x,y)
rry
rx
j i
sin
cos
yxr
ryr
rxr
y
x
222 zyxr
Physics 201: Lecture 4, Pg 24
Motion in 2 or 3 dimensionsPosition
Displacement
Velocity (avg.)
Acceleration (avg.)
ffii trtr , and ,
if rrr
t
r
avg.v
t v
a avg.