AP Physics Chapter 2

1
AP Physics Chapter 2

• Motion Along a Straight Line

2
AP Physics

• Turn in Chapter 1 Homework, Worksheet, Lab
• Take quiz
• Lecture
• QA

3
Motion

• Kinematics Study of motion, emphasizing on
  describing motion
• Simplest motion

• 1-Dimensional
• Point Particle no rotation

4
Position

• Position Location, where an object is
• Symbol x
• Unit meter
• How to describe a position in 1-D?

• Frame of Reference (numbered line)

• Reference point origin
• Direction

• Positive direction We can define the positive
  direction to be any direction we want, normally
  direction of motion
• Negative direction Opposite to positive direction

5
Position is a ______
vector

• Direction negative sign indicates direction only
  (more later)
• Magnitude how far from the origin

• More on Chapter 3

6
Position and Frame of Reference
East
West
-10
(- indicates only direction, west.)
10
0
7
So, Representing Position

• The representation of a position depends on the
  choice of the frame of reference.
• The same position can be expressed in different
  ways in different frame of reference.
• But the physical meaning of position does not
  change, regardless the choice of frame of
  reference.

• Hami still is 10 miles west of Downtown LA, no
  more or less.
• If you want to go from Downtown LA to Hami, you
  still have to go 10 miles west.

8
Displacement

• Displacement Change of position
• Symbol ?x, ? change
• ?x x2 - x1 xf xi
• Unit same as position, meter
• Displacement is also a vector

• Magnitude how far
• Direction Negative sign indicates direction
  only, it has nothing to do with magnitude.

larger than
A 3m displacement is ___________ a 2m
displacement.

• Displacement has nothing to do with the actual
  path. It depends only on the initial and final
  positions.

9
Distance (d)

• Distance ? Displacement
• Distance is a scalar. (magnitude only)
• Distance is equal to Magnitude of Displacement
  when there is no change in direction.
• d ?x
• Distance is not necessarily equal to the
  magnitude of total displacement.
• Displacement cares only end points distance
  cares both end points and the actual path.
• Representations of position and displacement
  depend on frame of reference, but distance does
  not depend on frame of reference.

10
Total Distance

• Whenever there is a change in direction, total
  distance will not be the same as the magnitude of
  total displacement.

When you go from x 0 to 3m then back to 2m,

• Your total displacement is __ m.
• Your total distance traveled is __ m.

2
4
11
Total Distance, dtot

• When there is no change in direction

• When there is change in direction

where d1 and d2 are distances of segments in
which there is no change in direction.
12
Average Velocity

• Average velocity ratio of displacement ?x that
  occurs during a particular time interval ?t to
  that time interval

• Standard Unit m/s

• Constant velocity

13
Average speed,

• Average speed is distance traveled during a time
  interval divided by the time interval.

14
Example During a hard sneeze, your eyes might
shut for 0.50 s. If you are driving a car at 90
km/h, how far does it move during that time?

• Given ?t 0.50s, v 90km/h
• ?x ?

15
Practice Boston Red Sox pitcher Roger Clemens
could routinely throw a fastball at a horizontal
speed of 160 km/h. How long did the ball take to
reach home plate 18.4 m away?

• Given

?t ?
16
Practice 30-11

• You drive on Interstate 10 from San Antonio to
  Houston, half the time at 55 km/h and the other
  half at 90 km/h. On the way back you travel half
  the distance at 55 km/h and the other half at 90
  km/h. What is your average speed
• from San Antonio to Houston,
• from Houston back to San Antonio, and
• for the entire trip?
• What is your average velocity for the entire
  trip?
• Graph x versus t for (a), assuming the motion is
  all in the positive x direction. Indicate how
  the average velocity can be found on the graph.

17
Practice

• a)
• Let T be the total number of hours from San
  Antonio to Houston, then the distance of the
  first T/2 hours is (55 ? T/2) km,
• and that of the second T/2 hours is (90 ? T/2)
  km.
• Total distance is (55 ? T/2) (90 ? T/2)
  72.5T km.
• Then average speed is

18
Practice (2)

• b)
• Let D km be the total distance. Then the time
  for the first D/2 km is

Similarly the time for the second D/2 km is D/180
h, then total time is
Then the average speed is
19
Practice (3)

• c)
• Similar to Part b), the average speed for the
  entire trip is

d) Total displacement is 0 for round trip. Then
average velocity is
20
Practice (4)
x
t
Average velocity can be found by finding the
slope of the line (in red) connecting the end
points.
21
Instantaneous velocity

• Instantaneous velocity is the average velocity
  when the time interval becomes very, very small,
  essentially zero.

• Instantaneous velocity is the time-derivative of
  position function.

22
Derivatives

• C is constant, and x and y are functions of t.

23
Try
24
Instantaneous velocity
What does the slope of this line mean?
Tangent line
Slope average velocity from time ti to t1
t4
ti
t2
t1
t3
As the time interval becomes smaller and smaller,
average velocity becomes instantaneous velocity,
which is the slope of the tangent line.
25
Derivative and slope

• The derivative of a curve at any point is the
  slope of its tangent line at that point.
• Instantaneous velocity v is the time derivative
  of position x. On a position-time graph, the
  instantaneous velocity at any time is the slope
  of the line tangent to the curve at that time.

• If position graph is a straight line

slope is _________, and
constant
_________ is constant.
velocity
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How far does the runner whose velocity-time graph
is shown below travel in 16s?

• Displacement Area under curve of v-t graph

v (m/s)
8
4
t (s)
0
16
12
10
27
A1
A1
28
Acceleration
Think of acceleration as a push as of now, though
not exactly correct.

• Average acceleration

• Instantaneous acceleration

29
Velocity-time graph

• On a velocity-time graph, the slope of the line
  tangent to the curve at any time is the
  instantaneous acceleration at that time.
• If the graph is a straight line, acceleration is
  constant.
• Displacement is area under curve

30
Position, Velocity, and acceleration Graphs

• Position (vs. Time) graph

v slope
?x Area under curve
Integrate
Derivative
Velocity (vs. Time) graph
a slope
?v Area under curve
Acceleration (vs. Time) graph
31
Speeding Up or Slowing down


? speeding up
? slowing down

-
-
-
? speeding up
? slowing down
-

• v and a are in the same direction (or have the
  same sign) ? ___________

speeding up

• v and a are in the opposite direction (or have
  the opposite signs) ? ____________

slowing down
32
Directions of acceleration and velocity

• Sign of a vector indicates direction only.
• If the signs of the velocity and acceleration of
  a particle are the same (same direction), the
  speed of the particle increases (speeding up).
• If the signs are opposite (opposite directions),
  the speed decrease (slowing down).

• A negative acceleration does not necessarily mean
  slowing down.
• Deceleration slow down

33
Speeding up or slowing down

• Speeding up

in the same

• Acceleration is ___________ direction of velocity.

• Then we can determine the sign of acceleration
  depending on what direction has been already
  defined as the positive direction

• Slowing down

opposite to

• Acceleration is __________ the direction of
  velocity.

34
Practice 35-79

• If the position of an object is given by x
  2.0t3, where x is measured in meters and t in
  seconds, find (a) the average velocity and (b)
  the average acceleration between t 1.0s and t
  2.0 s. Then find (c) the instantaneous
  velocities and (d) the instantaneous
  accelerations at t 1.0s and t 2.0 s. (e)
  Compare the average and instantaneous quantities
  and in each case explain why the larger one is
  larger. (f) Graph x versus t and v versus t, and
  indicate on the graphs your answers to (a)
  through (d).

35
Practice (2)

• a)

36
Practice (3)

• b)

37
Practice (4)

• c)

d)
38
Practice (5)

• d)

Because both the velocity and acceleration are
increasing, so the final values are the largest
ones.
39
Practice (6)
f)
a) Slope of red line is the average velocity from
1s to 2s
c) Slope of blue line is the instantaneous
velocity at 1 s
c) Slope of green line is the instantaneous
velocity at 2 s
40
Practice (7)
b) Slope of red line is the average acceleration
from 1s to 2s
d) Slope of blue line is the instantaneous
acceleration at t 1s
d) Slope of green line is the instantaneous
acceleration at t 2s
41
Constant Acceleration Motion

• a constant
• Let initial time t 0, then at any time t,
  velocity and position are given by

42
Free-Fall Motion

• Assume no air resistance. (Valid when speed is
  not too fast.)
• a g, downward (g 9.81 m/s2)

• Acceleration can be positive or negative,
  depending on what we define as the positive
  direction.
• g is always a positive number, equivalent to 9.81
  m/s2.
• Does not matter if the object is on its way up,
  on its way down, or at the very top.

• g is acceleration due to gravity (It is not
  gravity.) g does not depend on mass of object.

43
Terms

• Released, drop

? vi 0 relative to hand

• If hand not moving, then vi 0.
• If hand moving, then vi ? 0

• a g, downward, regardless of direction of
  velocity

• on the way up
• on the way down, or
• at the very top

? V 0
44
Signs of v and a
v
a
-
0
v
a

0
v
a

-
v
a
-

v
a


v
a
-
-
Define up
Define down
45
Free-Fall motion equations

• These equation are valid only when downward is
  defined as the positive direction.
• Not valid when upward is defined as the positive
  direction. (Must replace every g with g.)
• No need to remember these equations.

46
Practice A jumbo jet must reach a speed of 360
km/h (225 mi/h) on the runway for takeoff. What
is the least constant acceleration needed for
takeoff from a 1.80 km runway?
47
Practice A stone is thrown down from a bridge
43.9 m above the water and splashes the water 2.0
s later. What is the initial speed of the stone?
With what speed the stone hit the water?
Let down . Then
48
Practice A rock is thrown straight upward with
an initial speed of 10 m/s from a window that is
20 m high. A) How much higher can it go? B)
How fast is it moving when it hits the ground?
Let up , x 0 at window.
49
Practice A rocket-driven sled running on a
straight, level track is used to investigate the
physiological effects of large accelerations on
humans. One such sled can attain a speed of 1600
km/h in 1.8 s starting from rest. Find (a) the
acceleration (assumed constant) in g units and
(b) the distance traveled.