Chapter2: Motion Along a Straight Chapter2: Motion Along a Straight Line Line OBJECTIVES:OBJECTIVES:

Displacement, Time , and Average VelocityDisplacement, Time , and Average Velocity

Consider the motion of a car below.Consider the motion of a car below.

At time At time tt11 = 1s = 1s , the car is at position , the car is at position xx11 = =19m19m and At time and At time tt22 = 4s= 4s , the car is at position , the car is at position xx2 2 = = 277m.277m.

The displacement of the car The displacement of the car is: is:

mmmxxx 2581927712

The time interval is: The time interval is: sssttt 31412

The average velocity of the car The average velocity of the car is:is: t

xvav

12

12

tt

xxvav

sms

mvav /86

3

258

Graphs of Motion Part 1: Position vs. TimeGraphs of Motion Part 1: Position vs. Time

The object is at The object is at rest. rest.

x (

m)

x (

m)

t (s)t (s)

x (

m)

x (

m)

t (s)t (s)

x (

m)

x (

m)

t (s)t (s)x (

m)

x (

m)

t (s)t (s)

x (

m)

x (

m)

t (s)t (s)

The object is moving The object is moving with constant with constant velocity in the velocity in the positive direction. positive direction.

The object is moving The object is moving with constant with constant velocity in the velocity in the negative direction. negative direction.

The object is moving The object is moving with increasing with increasing velocity velocity (accelerating). (accelerating).

The object is moving The object is moving with decreasing with decreasing velocity velocity (decelerating). (decelerating).

EXAMPLE 1:EXAMPLE 1:

smss

mmmv

smm

smmave /4.4

7040

480)280()200(

/4280

/5200

(a) average (a) average speedspeed

(b) average velocity(b) average velocity

smss

mmmv

smm

smmave /73.0

7040

80)280()200(

/4280

/5200

Instantaneous VelocityInstantaneous Velocity

instantaneous velocity (v)instantaneous velocity (v) is the velocity of a body at a specific is the velocity of a body at a specific instant of time or at any specific point along the path. instant of time or at any specific point along the path.

instantaneous velocity (v)instantaneous velocity (v) is the limit of average velocity of a is the limit of average velocity of a body as the time interval approaches zero. body as the time interval approaches zero.

t

xtv

t

0lim)(

instantaneous velocity (v)instantaneous velocity (v) is the derivative of displacement is the derivative of displacement with respect to time. with respect to time.

dt

dxtv )(

EXAMPLE EXAMPLE 2:2:

3322 )/120.0()/40.2()( tsmtsmtx (a)(a)

0)0)(120.0()0)(40.2()0( 32 x

mx 120)10)(120.0()10)(40.2()10( 32

smm

m

tt

xxvave /12

010

0120

12

12

EXAMPLE EXAMPLE 2:2:

23232 )/360.0()/80.4()120.040.2()( tsmtsmttdt

d

dt

dxtv (b)(b)

0)0)(360.0()0)(80.4()0( 2 v

smv /33)5)(360.0()5)(80.4()5( 2

smv /12)10)(360.0()10)(80.4()10( 2

(c)(c) 2360.080.40 tt st 3.13

Average and Instantaneous AccelerationAverage and Instantaneous Acceleration

acceleration (a)acceleration (a) is the time rate of change in velocity. is the time rate of change in velocity.

average acceleration average acceleration (a(aavav)) 12

12

tt

vv

t

vaav

instantaneous acceleration (a)instantaneous acceleration (a) is the limit of average is the limit of average acceleration as the time interval approaches zero. acceleration as the time interval approaches zero.

dt

dvta )(t

vta

t

0lim)(

23)/100.0()/00.3()( tsmsmtv (a)(a)

EXAMPLE EXAMPLE 3:3:

smsmsmv /00.3)0)(/100.0()/00.3()0( 23

smssmsmv /500.3)5)(/100.0()/00.3()5( 23 2

12

12 /500.0)05(

)/500.0()/00.3(sm

s

smsm

tt

vvaave

tsmta )/200.0()( 3(b)(b) 0)0)(/200.0()0( 3 sma

23 /00.1)5)(/200.0()5( smssma

EXAMPLE EXAMPLE 3:3:

(c)(c)

Graphs of Motion Part 2: Velocity vs. TimeGraphs of Motion Part 2: Velocity vs. Time

The object is at The object is at rest. rest.

v (

m/s

)v (

m/s

)

t (s)t (s)

v (

m/s

)v (

m/s

)

t (s)t (s)v (

m/s

)v (

m/s

)

t (s)t (s)

v

(m/s

)v

(m/s

)

t (s)t (s)

The object is moving The object is moving with constant with constant velocity in the velocity in the positive direction. positive direction.

The object is moving The object is moving with increasing with increasing velocity (uniformly velocity (uniformly accelerating). accelerating).

The object is moving The object is moving with decreasing with decreasing velocity (uniformly velocity (uniformly decelerating). decelerating).

v v

(m/s

)(m

/s)

The object is moving The object is moving with constant with constant velocity in the velocity in the negative direction. negative direction.

t (s)t (s)

? Graphs of Motion Part 3: Acceleration vs. ? Graphs of Motion Part 3: Acceleration vs. TimeTime

a a

(m/s

(m/s

22))

t (s)t (s)

The object is moving The object is moving with constant with constant velocity in the velocity in the positive direction. positive direction.

The object is moving The object is moving with constant with constant velocity in the velocity in the negative direction. negative direction.

a a

(m/s

(m/s

22))

t (s)t (s)

The object is moving The object is moving with increasing with increasing velocity (uniformly velocity (uniformly accelerating). accelerating).

a a

(m/s

(m/s

22))

t (s)t (s)

The object is moving The object is moving with decreasing with decreasing velocity (uniformly velocity (uniformly decelerating). decelerating).

a a

(m/s

(m/s

22))

t (s)t (s)

Uniformly Accelerated Motion (UAM) Uniformly Accelerated Motion (UAM)

uniformly accelerated motionuniformly accelerated motion is motion with constant is motion with constant acceleration.acceleration.

t

xvav

Equations of Uniformly Accelerated Equations of Uniformly Accelerated Motion Motion

2

vvv oav

t

vva o

Equation 1: Equation 1:

Equation 2: Equation 2:

Equation 3: Equation 3:

assuming that tassuming that t00 = 0. = 0.

assuming that tassuming that t00 = 0. = 0.

dt

dxv

Equations of Uniformly Accelerated Equations of Uniformly Accelerated Motion Motion

Recall: Recall:

Equation 4: Equation 4:

Equation 5: Equation 5:

dtvdx t

t o

t

t

x

x ooo

dtatvvdtdx )(

221 attvxx oo 2

21 attvx o

dt

dva Recall: Recall: dtadv adxdta

dt

dxvdv

v

v

x

x oo

vdva

dx1

22

1 22o

o

vv

axx

a

vvx o

2

22

(a)(a)

EXAMPLE EXAMPLE 4:4:

GIVEN: GIVEN: x = 70.0m ; t = 7.00 s ; v = 15.0 m/sx = 70.0m ; t = 7.00 s ; v = 15.0 m/sFIND: (a) vFIND: (a) voo and (b) a and (b) a

sms

m

t

xvav /0.10

7

70

sm

vsmvvv ooav /0.10

2

/15

2

smsmsmvo /00.5)/15()/10(2

(b)(b) 2/43.17

)/5()/15(sm

s

smsm

t

vva o

EXAMPLE EXAMPLE 5:5:

EXAMPLE EXAMPLE 5:5:

Free Fall (UAM along the y-axis) Free Fall (UAM along the y-axis)

free fall free fall is motion under the action of the force of gravity alone is motion under the action of the force of gravity alone (air resistance is neglected).(air resistance is neglected).

• a freely-falling body has a constant acceleration called the a freely-falling body has a constant acceleration called the acceleration due to gravityacceleration due to gravity g = - 9.80 m/sg = - 9.80 m/s22 (always directed (always directed downward).downward).

t

yvav

Equations of Free Equations of Free FallFall

2

vvv oav

t

vvg o

Equation 1: Equation 1:

Equation 2: Equation 2:

Equation 3: Equation 3:

221 gttvy o

g

vvy o

2

22

Equation 4: Equation 4:

Equation 5: Equation 5:

NOTE: Follow correct sign NOTE: Follow correct sign convention. convention. All quantities with All quantities with downward direction should downward direction should have a negative sign.have a negative sign.

downward velocity: - vdownward velocity: - vupward velocity: +vupward velocity: +vdownward displacement: - downward displacement: - yyupward displacement: + upward displacement: + yyacceleration: g= -9.80 m/sacceleration: g= -9.80 m/s22

EXAMPLE EXAMPLE 6:6:

EXAMPLE EXAMPLE 7:7:

ANSWERS:ANSWERS:

(a) (a) yy11 = 10.1m; v = 10.1m; v11 = 5.2 m/s and y = 5.2 m/s and y44 = -18.4 m; v = -18.4 m; v11 =-24.2 m/s =-24.2 m/s

(b) (b) v =±11.3 m/sv =±11.3 m/s

(c) (c) y =+11.5 my =+11.5 m

(d) (d) a = g = - 9.8 m/sa = g = - 9.8 m/s22

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