Physics 207: Lecture 2 Notes
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Transcript Physics 207: Lecture 2 Notes
Lecture 4
Goals for Chapter 3 & 4
Perform vector algebra
• (addition & subtraction) graphically or by xyz components
• Interconvert between Cartesian and Polar coordinates
Work with 2D motion
• Distinguish position-time graphs from particle trajectory plots
• Trajectories
Obtain velocities
Acceleration: Deduce components parallel and
perpendicular to the trajectory path
Solve classic problems with acceleration(s) in 2D
(including linear, projectile and circular motion)
Discern different reference frames and understand how
they relate to motion in stationary and moving frames
Assignment: Read thru Chapter 5.4
MP Problem Set 2 due this Wednesday
Physics 207: Lecture 4, Pg 1
Example of a 1D motion problem
A cart is initially traveling East at a constant speed of
20 m/s. When it is halfway (in distance) to its destination
its speed suddenly increases and thereafter remains
constant. All told the cart spends a total of 10 s in transit
with an average speed of 25 m/s.
What is the speed of the cart during the 2nd half of the trip?
Dynamical relationships (only if constant acceleration):
x x0 v x t a x t
0
v x v x a x t
0
a x const
And
1
2
2
v 2x v x2 2a x (x x 0 )
0
v x (avg)
1
(v x v x )
2
0
x(displaceme nt )
vaverage velocity
t ( total time )
Physics 207: Lecture 4, Pg 2
The picture
x0
t0
v0
v1 ( > v0 )
a0=0 m/s2
a1=0 m/s2
x1 t1
x
x x2 x0
v
t t2 t0
t2
2
Plus the average velocity
Knowns:
x0 = 0 m
t0 = 0 s
0
x
v0 = 20 m/s
vx vx
t2 = 10 s
vavg = 25 m/s
ax 0
relationship between x1 and x2
Four unknowns x1 v1 t1 & x2 and must find v1 in terms of knowns
x x v t
0
0
Physics 207: Lecture 4, Pg 3
x x0 v x t
Using
0
x0
v0
v1 ( > v0 )
a0=0 m/s2
a1=0 m/s2
t0
x1 x0 v0 (t1 t0 )
Four
unknowns
Four
relationships
x1 t1
x
x2 x1 v1 (t2 t1 )
t2
2
x x2 x0
v
t t2 t0
x1 ( x2 x0 )
1
2
Physics 207: Lecture 4, Pg 4
x0 0
Using
x0
v0
v1 ( > v0 )
a0=0 m/s2
a1=0 m/s2
t0
x1 t1
1
t0 0
x1 v0 t1
Eliminate
unknowns
3
x2 x1 v1 (t2 t1 )
x1 x2
1
2
4
x2 12 x2 v1 (t2 t1 )
first x1
2&3
next t1
2
x
1
1
2
x2 v1 (t2 )
4
1
2
vt2 v1 (t2 )
then x2
x2
v0
t2
2
x2
v
t2
1
2
1
2
vt2
v0
Mult. by 2/ t2
v v1 (2 vv )
0
Physics 207: Lecture 4, Pg 5
Fini
x0
v0
v1 ( > v0 )
a0=0 m/s2
a1=0 m/s2
t0
Plus the average velocity
Given:
v0 = 20 m/s
t2 = 10 s
vavg = 25 m/s
x1 t1
x
v v1 (2 vv )
t2
2
0
v1
v v0
2 v0 v
v1 225 20m/sm/s2025m/sm/s
50015m/s
33.3 m/s
Physics 207: Lecture 4, Pg 6
Vectors and 2D vector addition
The sum of two vectors is another vector.
B
A =B+C
B
A
C
C
D = B + 2 C ???
Physics 207: Lecture 4, Pg 7
2D Vector subtraction
Vector subtraction can be defined in terms of addition.
B-C
= B + (-1)C
B
B-C
-C
C
B
A =B+C
A
Different direction
and magnitude !
Physics 207: Lecture 4, Pg 8
Reference vectors: Unit Vectors
A Unit Vector is a vector having length 1
and no units
It is used to specify a direction.
Unit vector u points in the direction of U
Often denoted with a “hat”: u = û
Useful examples are the
cartesian unit vectors [ i, j, k ]
Point in the direction of the
x, y and z axes.
R = rx i + ry j + rz k
U = |U| û
û
y
j
k
i
x
z
Physics 207: Lecture 4, Pg 9
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 207: Lecture 4, Pg 10
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 207: Lecture 4, Pg 11
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 207: Lecture 4, Pg 12
Converting Coordinate Systems
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
ry y r cos q
R x ˆi y ˆj
r x2 y 2
qtan-1 ( y / x )
y
ry
(x,y)
r
q
rx
x
• In 3D cylindrical coordinates (r,q,z), r is the same as the
magnitude of the vector in the x-y plane [sqrt(x2 +y2)]
Physics 207: Lecture 4, Pg 13
Resolving vectors into components
A mass on a frictionless inclined plane
A block of mass m slides down a frictionless ramp
that makes angle q with respect to horizontal.
What is its acceleration a ?
m
a
q
Physics 207: Lecture 4, Pg 14
Resolving vectors, little g & the inclined plane
gq
y
q
x
g (bold face, vector) can be resolved into its x,y or x’,y’
components
g=-gj
g = - g cos q j’ + g sin q i’
The bigger the tilt the faster the acceleration…..
along the incline
Physics 207: Lecture 4, Pg 15
Dynamics II: Motion along a line but with a twist
(2D dimensional motion, magnitude and directions)
Particle motions involve a path or trajectory
Recall instantaneous velocity and acceleration
These are vector expressions reflecting x, y & z motion
r = r(t)
v = dr / dt
a = d2r / dt2
Physics 207: Lecture 4, Pg 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.
The magnitude of the
instantaneous velocity
vector is the speed, s.
(Knight uses v)
s = (vx2 + vy2 + vz )1/2
v
Physics 207: Lecture 4, Pg 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).
The average
acceleration is a
vector quantity
directed along ∆v
( a vector! )
a
Physics 207: Lecture 4, Pg 18
Instantaneous Acceleration
The instantaneous acceleration is the limit of the average
acceleration as ∆v/∆t approaches zero
The instantaneous acceleration is a vector with components
parallel (tangential) and/or perpendicular (radial) to the
tangent of the path
Changes in a particle’s path may produce an acceleration
The magnitude of the velocity vector may change
The direction of the velocity vector may change
(Even if the magnitude remains constant)
Both may change simultaneously (depends: path vs time)
Physics 207: Lecture 4, Pg 19
Generalized motion with non-zero acceleration:
at a||
ar a
v
2
2
a 0 with a ar at
a
need both path & time
a = a + a
Two possible options:
Change in the magnitude of v
a
=0
v
a
=0
Change in the direction of
Animation
Physics 207: Lecture 4, Pg 20
Kinematics
The position, velocity, and acceleration of a particle in
3-dimensions can be expressed as:
r= xi +y j+z k
v = v x i + v y j + vz k
a = a x i + ay j + a z k
(i , j , k unit vectors )
x x(t )
y y (t )
z z (t )
dx
vx
dt
d 2x
ax 2
dt
dy
vy
dt
d2y
ay 2
dt
dz
vz
dt
d 2z
az 2
dt
with, if constant accel., e.g. x(t ) x0 vx t 12 ax t 2
0
All this complexity is hidden away in
r = r(t)
v = dr / dt
a = d2r / dt2
Physics 207: Lecture 4, Pg 21
Special Case
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 t
x
y
y vs t
t=0
y
0
4
t
0
4
t
x vs y
4
x
Physics 207: Lecture 4, Pg 22
Another trajectory
Can you identify the dynamics in this picture?
How many distinct regimes are there?
Are vx or vy = 0 ? Is vx >,< or = vy ?
t=0
x vs y
y
t =10
x
Physics 207: Lecture 4, Pg 23
Another trajectory
Can you identify the dynamics in this picture?
How many distinct regimes are there?
0<t<3
I.
3<t<7
7 < t < 10
vx = constant = v0 ; vy = 0
II. vx = vy = v0
t=0
III. vx = 0 ; vy = constant < v0
x vs y
What can you say about the
acceleration?
y
t =10
x
Physics 207: Lecture 4, Pg 24
Exercise 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.
Sketch the shape of the path
after point C
Physics 207: Lecture 4, Pg 25
Exercise 1
Trajectories with acceleration
From B to C ?
A.
B.
C.
D.
E.
A
B
C
D
None of these
B
A
B
C
B
B
C
C
B
C
D
C
Physics 207: Lecture 4, Pg 26
Exercise 3
Trajectories with acceleration
After C ?
A.
B.
C.
D.
E.
A
B
C
D
None of these
C
C
A
B
C
C
C
D
Physics 207: Lecture 4, Pg 27
Lecture 4
Assignment: Read through Chapter 5.4
MP Problem Set 2 due Wednesday
Physics 207: Lecture 4, Pg 28