Motion in One Dimension Physics 2053 Lecture Notes 02a dx dt x t Kinematics in One Dimension (Phy 2053) vittitoe

Motion in One Dimension

Physics 2053Lecture Notes 02a

dx

dt

x

t

Kinematics in One Dimension (Phy 2053) vittitoe

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


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


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


(m)

x = x xo

= 6 m 6 m

= 12 m

Note: Displacement to the left is negative

Displacement


0 2 4 6246

x


(m)

x = x xo

= ( m) (6 m)

= 8 m

Note: Displacement to the right is positive

Displacement


Displacement


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


Average speed

The average speed of a particle is defined as

time totaldistance total

speed Average

Velocity


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


Instantaneous speed

The instantaneous speed of a particle is defined as the magnitude of its instantaneous velocity.

Velocity


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


Acceleration


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


Acceleration


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.


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



(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)



(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.




(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.




(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.



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.


(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


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


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


Motion Diagrams


The slope of a position versus time graph gives

A) position.

B) velocity.

C) acceleration.

D) displacement.

Motion Diagrams


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


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



1 Eq atvv o

atvv o

atv

What are we calculating?

0 t

a

V





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



What are we calculating?

2 Eq 2

attvx

2

o

tvo

2at2

0 t

vo

v


att av




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





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


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?


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?


t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

Alternate Solutions

Problem


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?


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


t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

Alternate Solutions

Problem


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


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


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


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


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


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


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


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


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


t

x

t

v

t

a

t

v

x

t

x

t

v

t

a

Review


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


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.


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.


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.


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Motion in One Dimension Physics 2053 Lecture Notes 02a dx dt x t Kinematics in One Dimension (Phy 2053) vittitoe