Transcript Physics 350 - Los Rios Community College District

Chapter 2
Motion in One Dimension

Classical Physics
 Refers to physics before the 1900’s
▪ Kinematics/Dynamics
▪ Electromagnetism
▪ Thermodynamics
 Applies to everyday phenomena

Modern Physics
 Refers to post 1900’s physics
▪ Quantum Mechanics
▪ Relativity
▪ Nuclear Physics
 Applies to very small/big phenomena

Kinematics (special branch of mechanics)
 Study of motion
 Irrespective of causes (dynamics)
 Three most important concepts
▪ Displacement
▪ Velocity
▪ Acceleration

Assumptions
 Ideal Particle
▪ Classical physics concept
▪ Point-like object, no size
▪ Real particles have size, charge, spin
 Time is absolute
▪ Independent of position or velocity
▪ Relativity says time is not absolute!
▪ Twin Paradox

Define frame of reference
 Coordinate system
▪ Rectangular
 Simplest case
▪ 1-Dimensional
▪ Can extrapolate to other dimensions (independent)
12 34 5678 9
X
 Only motion along the straight line is possible

Allows keep track of direction
 Positive direction
 Negative direction

When you apply equations for the motion of
bodies it is very important to keep track of
the direction of motion
 Negative and Positive values
 Affect result
The displacement of a moving object moving
along the x-axis is defined as the change in
the position of the object,
Δx = xf – xi
Where xi is the initial position and xf is the final
position
First Displacement
Δx1 = xf – xi = 52 – 30 = 22 m
Second Displacement
Δx2 = xf – xi = 38 – 52 = -14 m can have negative displacement – backwards

Vector – both magnitude and direction
 Ex: Velocity – magnitude (speed) and direction
N
▪ 55 mi/hr North East
NE
▪ 45º North of East
E
 Ex: Electric Field – magnitude and direction
▪ Electron
▪ Perpendicular to surface

Scalar – magnitude only
 Ex: temperature, density, mass, volume

Think of a vector as an arrow (pointing to
some direction), and its magnitude as the
length of the arrow (always positive,
independent of the direction of the arrow).
ConcepTest
Texting While Walking
You and your friend go for a walk to the park.
On the way, your friend decides to text while
walking and wanders off and takes a few side
trips by dodging cars and falling off a bridge.
When you both arrive at the park, do you and
your friend have the same displacement?
1) yes
2) no
ConcepTest
Texting While Walking
You and your dog go for a walk to the park. On
the way, your dog takes many side trips to
chase squirrels or examine fire hydrants.
When you arrive at the park, do you and your
dog have the same displacement?
1) yes
2) no
Yes, you have the same displacement. Since you and your friend
had the same initial position and the same final position, then
you have (by definition) the same displacement.
Follow-up: Have you and your friend traveled the same distance?


The average speed of an object is given by:
Average speed = total distance / total time
s = d / t > 0 always
Speeds (m/s):
▪
▪
▪
▪
▪
Light
3 x 108
Sound
343
Person
10
Fastest Car 110
Continent 10-8

Speed and motion are relative
 Depends on frame of reference
A person running on earth - 5 m/s
Sun
Earth
Orbit speed of earth around
the sun - 29.7 km/s

The average velocity v during a time interval
Δt is the displacement Δx divided by Δt :
f - xi
v = Δx
— = x——
Δt
tf - ti
(m/s)
 One dimension, straight line motion

The average velocity is equal to the slope of
the straight line joining the initial and final
points on a graph of the position vs. time

The figure below shows the graphical interpretation of the
average velocity (A to B), in the case of an object moving
with a variable velocity:

Example:
A motorist drives north for 35.0 minutes at
85.0 km/h and then stops for 15.0 minutes.
He then continues north, traveling 130 km in
2.00 h. (a) What is his total displacement? (b)
What is his average velocity?

The slope of the line tangent to the position vs. time
curve at some point is equal to the instantaneous
velocity at that time.
Δx
v = lim ——
Δx -> 0 Δt
x
t
Position vs. Time graphs
An object moving with a constant
velocity will have a graph that is a
straight line
An object moving with a non-constant
velocity will have a graph that is a
curved line

Position vs. Time
 Time is always moving forward
Example:
a) What is the velocity from O to A?
b) What is the velocity from A to B?
c) What is the velocity from O to C?
d) What is the instantaneous velocity at
t=2?

Note: Average velocity does not necessarily
have the same magnitude as average speed
Distance Travelled x3
Average speed = ————————
Time
Displacement
Average velocity = ————————
Time
x1
x2 - x1
s12 = —————— ≠ 0
t2 - t1
x2 - x1
v12 = —————— = 0
t2 - t1
x2
t1
t2
1.
A yellow car is heading East at 100 km/h
and a red car is going North at 100 km/h.
Do they have the same speed? Do they
have the same velocity?
2.
A 16-lb bowling ball in a bowling alley in
Folsom Lanes heads due north at 10 m/s.
At the same time, a purple 8-lb ball heads
due north at 10 m/s in an alley in San
Francisco. Do they have the same
velocity?
• The average acceleration v of an object with a
change of velocity Δv during a time interval Δt :
Δv
vf - vi
Δt
tf - ti
a = — = ——
(m/s2)
• The instantaneous acceleration of an
object at a certain time equals the slope of a
velocity vs. time graph at that instant.
Average
Acceleration

Acceleration is a vector quantity
 It has both a magnitude and a direction.



Positive or negative to indicate direction (of
acceleration!)
Acceleration is positive when the velocity
increases in the positive direction
Furthermore, the velocity increases when the
acceleration and the velocity point in the
same direction, it decreases when they are
pointing in opposite directions

Positive Acceleration

Negative Acceleration

Positive Acceleration (Negative direction)
If the velocity of a car is non-
1) yes
zero (v  0), can the
2) no
acceleration of the car be zero?
3) depends on
the velocity
1) yes
If the velocity of a car is non-
2) no
zero (v 0), can the
3) depends on the
acceleration of the car be zero?
velocity
Sure it can! An object moving with constant velocity has a
non-zero velocity, but it has zero acceleration since the
velocity is not changing.
Velocity vs. Time graphs
Velocity vs. Time graphs
Velocity vs. Time graphs

Example:
A car traveling in a straight line has a velocity
of + 5.0 m/s at some instant. After 4.0 s, its
velocity is + 8.0 m/s. What is the car’s
average acceleration during the 4.0-s time
interval?

Useful to look at situations when
 the acceleration is constant
▪ Velocity is changing
 Motion in a straight line

If the acceleration is constant, then the
average acceleration is equal to the
acceleration itself:
 a = constant => <a> = a = constant

Applies to gravity and other situations as well

Important (and useful) equations with straight line, uniform
acceleration
v = v0 + at
2. Δx = vt = ½(v0 + v)t
3. Δx = v0t + ½ at2
4. v2= v02 + 2a Δx
1.

ALL you need to solve problems of kinematics
with constant acceleration in 1 dimension

Examples

The most important example of 1d motion
with uniform acceleration is gravity

A free-falling object is an object falling under
the influence of gravity alone


All objects fall near the earth’s surface with a
constant acceleration, g
g = 9.8 m/s2
 g is always directed downward

All objects, regardless of mass, free-fall at the
same acceleration

An example of free fall is the collapse of gas in the
formation of stars

Gravitational Force
 Dependent on where we are on Earth, because the Earth is
not a perfect sphere and because the presence of large
masses (e.g. mountains) also affects the local gravitational
force
 For now, just consider the force of gravity as a constant
force pulling any object towards the center of the Earth,
and therefore perpendicular to the ground and towards
the ground.
▪ Approximate constant acceleration pointing to the ground

Feather and Hammer
 http://www.youtube.com/watch?v=KDp1tiUsZw8

Falling objects
 http://www.youtube.com/watch?v=_XJcZ-KoL9o
ConcepTest 2.9b
Alice and Bill are at the top of a
building. Alice throws her ball
downward. Bill simply drops
his ball. Which ball has the
greater acceleration just after
release?
Free Fall II
1) Alice’s ball
2) it depends on how hard
the ball was thrown
3) neither -- they both have
the same acceleration
4) Bill’s ball
Alice
v
v
0
A
Bill
v
B
ConcepTest 2.9b
Alice and Bill are at the top of a
building. Alice throws her ball
downward. Bill simply drops
his ball. Which ball has the
greater acceleration just after
release?
Both balls are in free fall once they are
released, therefore they both feel the
acceleration due to gravity (g). This
acceleration is independent of the initial
velocity of the ball.
Free Fall II
1) Alice’s ball
2) it depends on how hard
the ball was thrown
3) neither -- they both have
the same acceleration
4) Bill’s ball
Alice
v
v
0
A
Follow-up: Which one has the greater velocity when they hit
the ground?
Bill
v
B

All objects, regardless of mass, free-fall at the
same acceleration
 More detail in chapters on force and mass
 Free-falling objects do not encounter air
resistance

We can use the same four important
equations from before (because g is constant
acceleration) but change x direction to y
direction
 Substitute a = -g
Substitute a = -g, y-direction
1. v = v0 - gt
2.
3.
4.
Δy = vt = ½(v0 + v)t
2
Δy = v0t - ½ gt
2
2
v = v0 - 2g Δy

Imagine dropping an object, and measuring how
fast it’s moving over consecutive 1 second intervals

http://www.youtube.com/watch?v=xQ4znShlK5A

The vertical component of velocity is changing by
9.8 m/s in each second, downwards

Let’s approximate this acceleration as 10 m/s2
Time
Interval
0–1s
Acceleration
(m/s2 down)
10
Vel. at end of
interval
(m/s down)
Starting from rest, then letting go.
After an interval t, the velocity
changes by an amount at, so that
10
vfinal = vinitial + at
1–2s
10
20
2–3s
10
30
3–4s
10
40
4–5s
10
50
How fast was it going at the end
of 3 sec?
vinitial was 20 m/s after 2 sec
a was 10 m/s (as always)
t was 1 sec (interval)
vfinal = 20 m/s + 10 m/s2  1 s
= 30 m/s
You throw a ball upward with an
1) more than 10 m/s
initial speed of 10 m/s. Assuming
2) 10 m/s
that there is no air resistance,
3) less than 10 m/s
what is its speed when it returns to
4) zero
you?
5) need more information
You throw a ball upward with an
initial speed of 10 m/s. Assuming
that there is no air resistance,
what is its speed when it returns to
1) more than 10 m/s
2) 10 m/s
3) less than 10 m/s
4) zero
5) need more information
you?
The ball is slowing down on the way up due to
gravity. Eventually it stops. Then it accelerates
downward due to gravity (again). Since a = g on
the way up and on the way down, the ball reaches
the same speed when it gets back to you as it had
when it left.
Example
a) Find the time when the stone
reaches its maximum height.
b) Determine the stone’s maximum
height.
c) Find the time the stone takes to
return to its final position and find the
velocity of the stone at that time.
d) Find the time required for the stone
to reach the ground.