Chapter 2 Motion in One Dimension. Classical Physics Refers to physics before the 1900’s ▪...
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Transcript of Chapter 2 Motion in One Dimension. Classical Physics Refers to physics before the 1900’s ▪...
Chapter 2Motion 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)
Only motion along the straight line is possible
1 2 3 4 5 6 7 8 9 X
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 DisplacementΔx1 = xf – xi = 52 – 30 = 22 mSecond DisplacementΔx2 = xf – xi = 38 – 52 = -14 m can have negative displacement – backwards
Vector – both magnitude and direction Ex: Velocity – magnitude (speed) and
direction▪ 55 mi/hr North East
▪ 45º North of East
Ex: Electric Field – magnitude and direction▪ Electron
▪ Perpendicular to surface
Scalar – magnitude only Ex: temperature, density, mass, volume
NE
E
N
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).
You and your friend go for a You and your friend go for a
walk to the park. On the way, walk to the park. On the way,
your friend decides to text while your friend decides to text while
walking and wanders off and takes walking and wanders off and takes
a few side trips by dodging cars a few side trips by dodging cars
and falling off a bridge. When you and falling off a bridge. When you
both arrive at the park, do you both arrive at the park, do you
and your friend have the same and your friend have the same
displacement?displacement?
1) yes
2) no
ConcepTest ConcepTest Texting While WalkingTexting While Walking
You and your dog go for a walk to You and your dog go for a walk to
the park. On the way, your dog the park. On the way, your dog
takes many side trips to chase takes many side trips to chase
squirrels or examine fire hydrants. squirrels or examine fire hydrants.
When you arrive at the park, do When you arrive at the park, do
you and your dog have the same you and your dog have the same
displacement?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.
ConcepTest ConcepTest Texting While WalkingTexting While Walking
Follow-up:Follow-up: Have you and your friend traveled the same Have you and your friend traveled the same distance?distance?
The average speed of an object is given by:
Average speed = total distance / total times = 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
SunEarth
Orbit speed of earth around the sun - 29.7 km/s
A person running on earth - 5 m/s
The average velocity v during a time interval Δt is the displacement Δx divided by Δt :
v = — = ——
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
tf - ti
xf - xi
Δt
Δx(m/s)
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.
t
x
v = lim ——Δx
ΔtΔx -> 0
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
Average speed = ————————
Average velocity = ————————
Distance Travelled
Time
Time
Displacement
x1
x3
t1 t2s12 = —————— ≠ 0
v12 = —————— = 0
x2
x2 - x1
t2 - t1 x2 - x1
t2 - t1
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 :
a = — = ——
• The instantaneous acceleration of an object at a certain time equals the slope of a velocity vs. time graph at that instant.
tf - ti
vf - vi
Δt
Δv(m/s2)
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-If the velocity of a car is non-
zero (zero (v v 00), can the ), can the
acceleration of the car be zero?acceleration of the car be zero?
1) yes
2) no
3)
depends
on the
velocity
Sure it can! An object moving with constantconstant
velocityvelocity has a non-zero velocity, but it has
zerozero accelerationacceleration since the velocity is not
changing.
If the velocity of a car is non-If the velocity of a car is non-
zero (zero (v v 00), can the acceleration ), can the acceleration
of the car be zero?of the car be zero?
1) yes
2) no
3) depends on the
velocity
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
1. v = v0 + at2. Δx = vt = ½(v0 + v)t3. Δx = v0t + ½ at2
4. v2= v02 + 2a Δx
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 downwardAll 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 Hammerhttp://www.youtube.com/watch?v=KDp1tiUsZw8
Falling objectshttp://www.youtube.com/watch?v=_XJcZ-KoL9o
ConcepTest 2.9bConcepTest 2.9b Free Fall II Free Fall II
Alice and Bill are at the top of a Alice and Bill are at the top of a
building. Alice throws her ball building. Alice throws her ball
downward. Bill simply drops downward. Bill simply drops
his ball. Which ball has the his ball. Which ball has the
greater acceleration just after greater acceleration just after
release?release?
1) Alice’s ball 1) Alice’s ball
2) it depends on how hard 2) it depends on how hard the ball was thrownthe ball was thrown
3) neither -- they both have 3) neither -- they both have the same accelerationthe same acceleration
4) Bill’s ball4) Bill’s ball
vv
00
BillBillAliceAlice
vv
AA
vv
BB
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.
ConcepTest 2.9bConcepTest 2.9b Free Fall II Free Fall II
Alice and Bill are at the top of a Alice and Bill are at the top of a
building. Alice throws her ball building. Alice throws her ball
downward. Bill simply drops downward. Bill simply drops
his ball. Which ball has the his ball. Which ball has the
greater acceleration just after greater acceleration just after
release?release?
1) Alice’s ball 1) Alice’s ball
2) it depends on how hard 2) it depends on how hard the ball was thrownthe ball was thrown
3) neither -- they both have 3) neither -- they both have the same accelerationthe same acceleration
4) Bill’s ball4) Bill’s ball
vv
00
BillBillAliceAlice
vv
AA
vv
BBFollow-up:Follow-up: Which one has the greater velocity when they hit Which one has the greater velocity when they hit the ground?the ground?
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 - gt2. Δy = vt = ½(v0 + v)t3. Δy = v0t - ½ gt2
4. v2= v02 - 2g Δy
Imagine dropping an object, and measuring
how fast it’s moving over consecutive 1
second intervals http://www.youtube.com/watch?v=xQ4znShl
K5A 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
TimeInterval
Acceleration(m/s2
down)
Vel. at end of interval(m/s down)
0 – 1 s
10 10
1 – 2 s
10 20
2 – 3 s
10 30
3 – 4 s
10 40
4 – 5 s
10 50
Starting from rest, then letting go.After an interval t, the velocity changes by an amount at, so that
vfinal = vinitial + at
How fast was it going at the end of 3 sec?vinitial was 20 m/s after 2 seca 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 You throw a ball upward with an
initial speed of 10 m/s. initial speed of 10 m/s.
Assuming that there is no air Assuming that there is no air
resistance, what is its speed resistance, what is its speed
when it returns to you?when it returns to you?
1) more than 10 m/s 1) more than 10 m/s
2) 10 m/s2) 10 m/s
3) less than 10 m/s3) less than 10 m/s
4) zero4) zero
5) need more information5) need more information
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.
You throw a ball upward with an You throw a ball upward with an
initial speed of 10 m/s. initial speed of 10 m/s.
Assuming that there is no air Assuming that there is no air
resistance, what is its speed resistance, what is its speed
when it returns to you?when it returns to you?
1) more than 10 m/s 1) more than 10 m/s
2) 10 m/s2) 10 m/s
3) less than 10 m/s3) less than 10 m/s
4) zero4) zero
5) need more information5) need more information
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.
Example