Motion in a straight line

• Average, and instantaneous velocity

• Average, and instantaneous acceleration

All found in University Physics, 2.2-2.4

Lecture 2

Today… weekendnotes.com

ReviewLast lecture we looked at :• The difference between a vector and a scalar quantity.

Vector notation.

• The distinction between distance and displacement. Distance is a scalar quantity. Displacement is a vector

• Addition and subtraction of vectors

• Components of vectors

• Unit vectors. The scalar (dot) product

Example- Adding vectors• Find A + B using the component method if A has

magnitude 5.0m at elevation of 600 and B - magnitude 3.0m at elevation of 300 as shown below.

mji

jmm

imm

jBAiBABAR yyxx

ˆ8.5ˆ1.5

ˆ)30sin0.360sin0.5(

ˆ)30cos0.360cos0.5(

ˆ)(ˆ)(

00

00

A

B

300

300

5m

3m

5.8m

5.1m

X

Y

011

22

7.481.5

8.5tantan

7.7)8.5()1.5(magnitudeResultant

m

m

R

RAngle

mmm

x

y

5.8m

5.1m

Example- Adding vectors (cont)

5.8m

5.1m

X X

Y Y

θ

Example- Calculation of components

5.8m

5.1m

X

Y

θ=120°

A

0

0

120sin

120cos

AA

AA

y

x

θ1=30° 0

0

30cos

30sin

AA

AA

y

x

θ1=30°

000000

000000

30sin90cos30cos90sin3090sin

30sin90sin30cos90cos3090cos

AAA

AAA

y

x

The Displacement vectorand

Distance scalar

If P2 is at P1 then displacement is zero but distance travelled still has a value

Average velocity =

displacement/time

Average speed =

distance/time

Speed and velocity

Speed is measured using distance and time:

Velocity is measured using displacement and time:

t

x

tt

xxvav

12

12

t

d

tt

dd

12

12speed

So speed is a scalar, velocity is a vector.For example 15 km/hour is a scalar. 58.4 m/s in a northerly direction is a velocity and is a vector.

Example - Average velocity vs speed Ian Thorpe’s personal best for 100 m freestyle in a 50 m pool in 2001 was 48.81 seconds.

His average SPEED would have been 2.05 m/s

His average VELOCITY however is zero m/s!webswimming.tripod.com

Average velocity

t

x

tt

xxav

12

12v

Average velocity is the displacement divided by the time interval

t

x

tt

xxav

12

12vx2=1680 mt2 = 85 sec

sms

m

t

xav /23

73

1675v

Example- Find the average velocity.

x1=5 mt1= 12 sec

stop

Only includes initial and final conditions, the average does not have details about the path or events. Such as stopping or drive over the mountain.

old.risk.ru

Average velocity from graph

Example - Average velocity

It takes Penny 37.0 seconds to drive along a 1.2 km airport runway, from a standing start at one end to the other end. What is her average velocity in completing this displacement.

To become a velocity this speed needs to include a direction.Average velocity v=32.4m/s in a due North direction.

hunternissan.com

Instantaneous velocity

t

x

tt

xxav

12

12vvelocityAverage

Instantaneous velocity is the limit of the average velocity as the time interval approaches zero. It is the instantaneous rate of change of position with time

dt

dx

t

x

t

lim

0instantv

howstuffworks.com

Example 2.1 Average and instantaneous velocity

mtx 2520

ischeetah ofPosition

Example 2.1Average and Instantaneous velocity (cont)

a) Find the displacement of the cheetah during the interval between t = 1.0 sec and t = 2.0 sec.

vav=Δx/Δt=(40 m– 25m) /(2s – 1s)=15 m/s in the positive x direction

b) Find the average velocity during this time interval.

mtx 2520

Example 2.1Average and Instantaneous velocity (cont)

sms

m

t

xSo

mxst

mxst

st

av /5.101.0

05.1v

05.26;1.1

;25;1

1.0

22

11

c) Find the instantaneous velocity at time t = 1.0 sec, by taking Δt = 0.1 sec, then Δt = 0.01 sec.

d) Derive a general expression for the instantaneous velocity as a function of time, and from it find v, at t = 1.0 sec and t = 2.0 sec.

sms

m

t

xSo

mxt

mxst

statst

av /05.1001.0

1005.0v

1005.25;s 01.1

25 ;1

1 01.0 Using

22

11

1 nn nttdt

d

mtx 2520

Instantaneous velocity from a graph

As P2 approaches P1 the slope of the line between P1 and P2 becomes the tangent at point P1 and its slope gives the value for instantaneous velocity at P1 ….. On a graph of position as a function of time for straight-line motion, the instantaneous velocity at any point is equal to the slope of the tangent to the curve at that point

Average acceleration

t

x

tt

xxav

12

12v

Average velocity is the displacement divided by the time interval

Average acceleration is the change in velocity divided by the time interval

tttaav

vvv

12

12

Units of acceleration are m/s2

Instantaneous acceleration

Instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero; it is the instantaneous rate of change of velocity with time

tttaav

vvv

onaccelarati Average12

12

2

2

0instant

vv

dt

xd

dt

dx

dt

d

dt

d

ta

t

lim

Example2.3- Instantaneous acceleration

smt /50.060v

by given iscar theof velocity The2

Example 2.3 Instantaneous acceleration (cont)

a) Find the change in velocity of the car in the time interval between t =1.0 sec and t = 3.0 sec.

At t = 1 sec..v = 60.5 m/s At t = 3sec..v = 64.5 m/s

b) Find the average acceleration in this time interval

smt /50.060v 2

Example 2.3 Instantaneous acceleration (cont)

c) Find the average acceleration at time t=1.0 s by taking Δt to be first 0.1 s, then 0.01 s.

2/05.11.0

5.60605.60v

1.0 using 1timeAt

sms

s

m

s

m

ta

stst

av

d) Derive an expression for the instantaneous acceleration at any time, and use it to find the acceleration at t = 1.0 s and t = 3.0 s.

2/005.101.0

5.6051005.60v

01.0using 1timeAt

sms

s

m

s

m

ta

stst

av

1 nn nttdt

d

smt /50.060v 2

Instantaneous acceleration from a graph

Review: What does the displacement vs time graph look like?

Review: What does the velocity vs time graph look like?

Example: A dizzy turkey• A turkey moves according to the following

relationship: x = 5 – 2 t + 0.5 t2 (m)• Calculate its instantaneous velocity and

acceleration at a time t = 5 s.

Differentiate: v = -2 + t (m/s); a = 1 m/s2

At t = 5 sv = 3 m/s; acceleration always 1m/s2

simpleist.com

Lets look at an object falling from afar.

An astronaut floats near the space shuttle. His acceleration is a function of his position.

wikipedia.org

legacy.shadowlordinc.com

However around the surface of the Earth the acceleration is fairly constant so a velocity time graph is a straight line.

vm/s

t sec

Remember…

• Average velocity=Δx/ Δt

• Instantaneous Velocity = dx/dt as Δt0….

… the derivative of x w.r.t t

• Average acceleration = Δv/ Δt

• Instantaneous acceleration = dv/dt as Δt0…. …the derivative of v w.r.t t

starwars.com

Lecture 2-What you need to know and where to find it

• Average, and instantaneous velocity

• Average, and instantaneous acceleration

All found in University Physics, 2.2-2.4