Motion in One Dimension – PART 2. 1624128402028 (s) x 4 8 12 16 20 24 28 (m) 1234 t 5 Motion Diagrams An object starts from rest and moves with constant.

Motion in One Dimension – PART 2

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

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.

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

Kinematics in One Dimension (Phy 2053) vittitoe

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

Kinematics in One Dimension (Phy 2053) vittitoe

(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

Motion Diagrams

Kinematics in One Dimension (Phy 2053) vittitoe

tx

v

The slope of a position versus time graph gives

A) position.

B) velocity.

C) acceleration.

D) displacement.

Motion Diagrams

Kinematics in One Dimension (Phy 2053) vittitoe

tv

a

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

Kinematics in One Dimension (Phy 2053) vittitoe

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

AB

B

A

v2v

at2t2av

atv

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

What are we calculating?

2 Eq 2

attvx

2

o

tvo

2at2

0 t

vo

v

One Dimensional Motion with Constant Acceleration

att av

One Dimensional Motion with Constant Acceleration

AB

22

oB

22

oA

x4x

2at

4 2t2a

t2vx

2at

2

attvx

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

One Dimensional Motion with Constant Acceleration

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?

)a v, ,v(t o

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

atvv o

2o m/s 0.2a 0,v m/s, 25v

a

vvt o 2m/s 0.2

m/s 25m/s 0

s 5.12

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 ,a v, ,v(x o

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

s 12.5t ,m/s 0.2a 0,v m/s, 25v 2o

t 2

vvx o

s 12.5 2

m/s 25m/s 0x

m 156

Alternate Solutions

Problem

)t ,a v, ,v(x o

xa2vv2

attvx

2o

2

2

o

s 12.5t ,m/s 0.2a 0,v m/s, 25v 2o

2at

tvx2

o

2

s 5.21m/s 0.2s 12.5m/s 25x

22

m 561x

xa2vv 2o

2

m 156

m/s 0.22

m/s 250a2

vvx

2

222o

2

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?

)x v, ,v(a o

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

m 196x 0,v m/s, 30vo

xa2vv 2o

2

x2

vva

2o

2

m 1962m/s 30m/s 0 22

2m/s 29.2

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

)a x, v, ,v(t o

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

2o m/s 2.29a m, 196x 0,v m/s, 30v

2m/s 29.2am 196x

atvv o

a

vvt o 2m/s 29.2

m/s 30m/s 0

s 1.13

Alternate Solutions

Problem

)a x, v, ,v(t o

t 2

vvx

xa2vv2

attvx

o

2o

2

2

o

2o m/s 2.29a m, 196x 0,v m/s, 30v

t 2

vvx o

s 1.13

m/s 30m/s 0m 1962

vvx2

to

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

)x v, ,v( a o m 240x km/h, 120v ,0vo

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

xa2vv 2

o2

x2

vva

2o

2

m 24020km/h 120 22

m 2402

0s 600,3

h 1km 1

m 000,1h

km 120

a

22

2m/s 31.2

A Cessna aircraft has a lift-off speed of 120 km/h.(b) How long does it take the aircraft to become airborne?

Problem

)a x, v, ,v( t o 2o m/s 2.31a m, 240x km/h, 120v ,0v

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

atvv o

a

vvt o

2m/s 31.2

0km/h 120

2m/s 31.2

0s 600,3

h 1km 1

m 000,1h

km 120

t

s 4.14

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

)a x, ,v( t o 2

o m/s 0.01a m, 400x ,0v

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

2at

tvx2

o

ax2

t

2m/s 01

m 4002 s 94.8

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 a, x, ,v( v o s948t,m/s 10.0a m, 400x ,0v 2o

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

atvv o

m/s 4.89 s .948m/s 100v

A ball is thrown vertically upward with a speed of 25.0 m/s. (a) How high does it rise?

Problem

)a v, ,v(x o2

o m/s 8.9ga 0,v m/s, 25v

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

xa2vv 2o

2

m 9.31x

a2

vvx

2o

2

2

22

m/s 8.92

m/s 250

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

)x a, v, ,v( t o m 31.9x ,m/s 8.9ga 0,v m/s, 25v 2o

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

atvv o

s 55.2t

a

vvt o 2m/s 8.9

m/s 250

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

)x a, ,v( t o m 31.9x ,m/s 8.9ga ,0v 2o

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

s 55.2t

2at

tvx2

o

ax2

t

2m/s 8.9

m/s 9.312

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

)x a, ,v( v o 0x ,m/s 8.9ga m/s, 25v 2o

t 2

vvx

xa2vv2

attvx

atvv

o

2o

2

2

o

o

xa2vv 2o

2

m/s 25v

xa2vv 2o

)0(m/s 8.92m/s 25v 22

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

Kinematics in One Dimension (Phy 2053) vittitoe

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