motion in one dimension physics 2053 lecture notes 02a dx dt x t kinematics in one dimension (phy...
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Motion in One Dimension
Physics 2053Lecture Notes 02a
dx
dt
x
t
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t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
2-01 Displacement
2-02 Velocity
2-03 Acceleration
2-04 Motion Diagrams
Motion in One Dimension
Sections
2-05 One Dimensional Motion with Constant Acceleration
2-06 Freely Falling Objects
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v
In the study of kinematics, we consider a moving object as a particle.
A particle is a point-like mass having infinitesimal size and a finite mass.
Displacement
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0 2 4 6246
x
Displacement
The displacement of a particle is defined as its change in position.
(m)
x = x xo
= 6 m 2 m
= 4 m
Note: Displacement to the right is positiveKinematics in One Dimension (Phy 2053) vittitoe
0 2 4 6246
x
The displacement of a particle is defined as its change in position.
(m)
x = x xo
= 6 m 6 m
= 12 m
Note: Displacement to the left is negative
Displacement
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0 2 4 6246
x
The displacement of a particle is defined as its change in position.
(m)
x = x xo
= ( m) (6 m)
= 8 m
Note: Displacement to the right is positive
Displacement
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Displacement
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A student walks 70 m East, then walks 30 km West.What is the magnitude of the students net displacement?
A) 30 m
B) 40 m
C) 70 m
D) 100 m
Average velocity
The average velocity of a particle is defined as xv
tx
vx
x
t
x1
x2
t1 t2
x
t
12
12ttxx
Velocity is represented by the slope on a
displacement-time graph
Velocity
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Average speed
The average speed of a particle is defined as
time totaldistance total
speed Average
Velocity
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Instantaneous velocity
The instantaneous velocity v, equals the limiting value of the ratio
t
xv lim
0tx
x
t
x
t
Instantaneous velocity is represented by
the slope of a displacement-time graph
Velocity
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Instantaneous speed
The instantaneous speed of a particle is defined as the magnitude of its instantaneous velocity.
Velocity
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Average acceleration
The average acceleration of a particle is defined
as the change in velocity vx divided by the time
interval t during which that change occurred.
tv
a xx
12
12ttvv
v
t
v1
v2
t1 t2
v
tAcceleration is
represented by the slope on a
velocity-time graph
Acceleration
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Acceleration
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A new car manufacturer advertises that their car can go "from zero to sixty in 8 s". This is a description of
A) instantaneous acceleration.
B) average speed.
C) instantaneous speed.
D) average acceleration.
Instantaneous acceleration
The instantaneous acceleration equals the derivative of the velocity with respect to time
t
va lim
0tx
v
t
v
t
Instantaneous acceleration is represented by
the slope of a velocity-time graph
Acceleration
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Acceleration
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A moving car experiences a constant acceleration of 1.5 m/s2. This means the car is
A) traveling at 1.5 m/s in every second.
B) changing its velocity by 1.5 m/s.
C) increasing its velocity by 1.5 m/s in every second.
D) increases its displacement by 1.50 m each second.
True or False?
(a) A car must always have an acceleration in the same direction as its velocity
Quick Quiz 2.2 (page 32)
(b) It’s possible for a slowing car to have a positive acceleration
(c) An object with constant nonzero acceleration can never stop and stay stopped.
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The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals
(a) 0 to 2 s
(b) 0 to 4 s
(c) 2 s to 4 s
(d) 4 s to 7 s
(e) 0 to 8 s.
Motion Diagrams
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The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals
(a) 0 to 2 s
(b) 0 to 4 s
(c) 2 s to 4 s
(d) 4 s to 7 s
(e) 0 to 8 s.
Motion Diagrams (con’t)
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The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals
(a) 0 to 2 s
(b) 0 to 4 s
(c) 2 s to 4 s
(d) 4 s to 7 s
(e) 0 to 8 s.
Motion Diagrams (con’t)
Kinematics in One Dimension (Phy 2053) vittitoe
The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals
(a) 0 to 2 s
(b) 0 to 4 s
(c) 2 s to 4 s
(d) 4 s to 7 s
(e) 0 to 8 s.
Motion Diagrams (con’t)
Kinematics in One Dimension (Phy 2053) vittitoe
The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals
(a) 0 to 2 s
(b) 0 to 4 s
(c) 2 s to 4 s
(d) 4 s to 7 s
(e) 0 to 8 s.
Motion Diagrams (con’t)
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16 2412840 20 28
(s)
x
4
8
12
16
20
24
28
(m)
1 2 3 4t
5
Motion Diagrams
An object starts from rest and moves with constant acceleration.
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(s)
x
48
1216202428
(m)
1 2 3 4 t5
1 2 3 4 5 t (s)
2
4
6
8
10
v
(m/s)sm
10v
s 5t
tv
a
s 5s
m10
2s
m 2
Displacement25 m
Motion Diagrams
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t
x
t
v
t
a
Displacement, velocityand acceleration graphs
The slope of a velocity-timegraph represents acceleration
tv
a
The slope of a displacement-timegraph represents velocity
tx
v
Motion Diagrams
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t
x
t
v
t
a
t
Displacement, velocityand acceleration graphs
The area under an acceleration-timegraph represents change in velocity.
v
vta
The area under a velocity-timegraph represents displacement.
x
xtv
Motion Diagrams
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Motion Diagrams
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The slope of a position versus time graph gives
A) position.
B) velocity.
C) acceleration.
D) displacement.
Motion Diagrams
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The slope of a velocity versus time graph gives
A) position.
B) velocity
C) acceleration
D) displacement
Definitions of velocity and acceleration
tx
vAverage velocity
tv
aAverage acceleration
One Dimensional Motion with Constant Acceleration
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For constant acceleration
An object moving with an initial velocity vo undergoes
a constant acceleration a for a time t. Find the final velocity.vo
time = 0 time = t
ΔtΔv
a Solution:
t avΔ atvv o Eq 1
a ?
atvv o
One Dimensional Motion with Constant Acceleration
Kinematics in One Dimension (Phy 2053) vittitoe
1 Eq atvv o
atvv o
atv
What are we calculating?
0 t
a
V
One Dimensional Motion with Constant Acceleration
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One Dimensional Motion with Constant Acceleration
Objects A and B both start at rest. They both accelerate at the same rate. However, object B accelerates for twice the time as object A. What is the final speed of object B compared to that of object A?
A) the same speed
B) twice as fast
C) three times as fast
D) four times as fast
For constant acceleration
An object moving with a velocity vo is passing position xo when it undergoes a constant acceleration a for a time t. Find the object’s displacement.
Solution:
time = 0 time = t
xo?
avo
tx
vavg
x2
attv
2
o xtvavg
x0t2
vvo
x0t2
atvv oo
2at
tvx2
o Eq 2
atvv o 1 Eq
One Dimensional Motion with Constant Acceleration
Kinematics in One Dimension (Phy 2053) vittitoe
What are we calculating?
2 Eq 2
attvx
2
o
tvo
2at2
0 t
vo
v
One Dimensional Motion with Constant Acceleration
att av
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One Dimensional Motion with Constant Acceleration
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Objects A and B both start at rest. They both accelerate at the same rate. However, object B accelerates for twice the time as object A. What is the distance traveled by object B compared to that of object A?
A) the same distance
B) twice as far
C) three times as far
D) four times as far
atvv o Eq 12
attvx
2
o Eq 2
Solve Eq 1 for a and sub into Eq 2:
2t
t
v-vtvx
2o
o
Solve Eq 1 for t and sub into Eq 2:
2oo
o a
vv
2a
a
vvvx
t 2
vvx o
Eq 3
xa2vv 2o
2 Eq 4
t
vva o
a
vvt o
One Dimensional Motion with Constant Acceleration
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One Dimensional Motion with Constant Acceleration
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When the velocity of an object is zero, must its acceleration also be zero?
A) no, an object thrown upward will have zero velocity at its highest point.
B) no, a falling object will have zero velocity after hitting the ground.
C) yes, if the object is not moving it can not be accelerating.
D) yes, acceleration implies a changing velocity, it can not be zero.
Freely Falling Objects
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When an object is released from rest and falls in the absence of air resistance, which of the following is true concerning its motion?
A) Its acceleration is constant
B) Its velocity is constant.
C) Neither its acceleration nor its velocity is constant.
D) Both its acceleration and its velocity are constant.
Problem
Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2. (a) How long does it take for the lead car to stop?
Kinematics in One Dimension (Phy 2053) vittitoe
t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
Problem
Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2. (b) How far does the lead car travel during the acceleration?
Kinematics in One Dimension (Phy 2053) vittitoe
t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
Alternate Solutions
Problem
Kinematics in One Dimension (Phy 2053) vittitoe
Problem
Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2.(c) Assuming that the chasing car brakes at the same time as the lead car, what must be the chasing car’s minimum negative acceleration so as not to hit the lead car?
Kinematics in One Dimension (Phy 2053) vittitoe
t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2.(d) How long does it take for the chasing car to stop?
Problem
2m/s 29.2am 196x
Kinematics in One Dimension (Phy 2053) vittitoe
t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
Alternate Solutions
Problem
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A Cessna aircraft has a lift-off speed of 120 km/h. (a) What minimum constant acceleration does the aircraft require if it is to be airborne after a takeoff run of 240 m?
Problem
Kinematics in One Dimension (Phy 2053) vittitoe
t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
A Cessna aircraft has a lift-off speed of 120 km/h.(b) How long does it take the aircraft to become airborne?
Problem
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t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
A drag racer starts her car from rest and accelerates at 10.0 m/s2 for a distance of 400 m (1/4 mile). (a) How long did it take the race car to travel this distance?
Problem
Kinematics in One Dimension (Phy 2053) vittitoe
t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
A drag racer starts her car from rest and accelerates
at 10.0 m/s2 for a distance of 400 m (1/4 mile). (b) What is the speed of the race car at the end of the run?
Problem
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t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
A ball is thrown vertically upward with a speed of 25.0 m/s. (a) How high does it rise?
Problem
Kinematics in One Dimension (Phy 2053) vittitoe
t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
A ball is thrown vertically upward with a speed of 25.0 m/s.(b) How long does it take to reach its highest point?
Problem
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t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
A ball is thrown vertically upward with a speed of 25.0 m/s.(c) How long does the ball take to hit the ground after it reaches its highest point?
Problem
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t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
A ball is thrown vertically upward with a speed of 25.0 m/s.(d) What is its velocity when it returns to the level from which it started?
Problem
t 2
vvx
xa2vv2
attvx
atvv
o
2o
2
2
o
o
Kinematics in One Dimension (Phy 2053) vittitoe
tx
vAverage velocity
tv
aAverage acceleration
atvv o 2
attvxx
2
oo
t 2
vvxx o
o
o2o
2 xxa2vv
Kinematics with Constant Acceleration
Definitions
Review
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t
x
t
v
t
a
t
v
x
t
x
t
v
t
a
Review
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Problem Solving Skills
1. Read the problem carefully
2. Sketch the problem
3. Visualize the physical situation
4. Identify the known and unknown quantities
5. Identify appropriate equations
6. Solve the equations
7. Check your answers
Review
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Suppose that an object is moving with a constant velocity. Make a statement concerning its acceleration.
A) The acceleration must be constantly increasing.
B) The acceleration must be constantly decreasing.
C) The acceleration must be a constant non-zero value.
D) The acceleration must be equal to zero.
Constant Velocity
Can an object have increasing speed while the magnitude of its acceleration is decreasing? Support your answer with an example.A) No, this is impossible because of the way in which acceleration is defined.
B) No, because if acceleration is decreasing the object will be slowing down.
C) Yes, and an example would be an object falling in the absence of air friction.
D) Yes, and an example would be an object released from rest in the presence of air friction.
Freely Falling Objects
Suppose a ball is thrown straight up. Make a statement about the velocity and the acceleration when the ball reaches the highest point.
A) Both its velocity and its acceleration are zero.
B) Its velocity is zero and its acceleration is not zero.
C) Its velocity is not zero and its acceleration is zero.
D) Neither its velocity nor its acceleration is zero.
Freely Falling Objects
Ball A is dropped from the top of a building. One second later, ball B is dropped from the same building. As time progresses, the distance between them
A) increases.
B) remains constant.
C) decreases.
D) cannot be determined from the information given.
Freely Falling Objects