Physics Beyond 2000

Chapter 1

Kinematics

Physical Quantities

• Fundamental quantities

• Derived quantities

Fundamental QuantitiesQuantity Symbol SI Unit

Mass m kg

Length l m

Time t s

Others - -

http://www.bipm.fr/

Derived Quantities

• Can be expressed in terms of the basic quantities

• Examples– Velocity– Example 1– Any suggestions?

Derived Quantities

• More examples

Standard Prefixes

• Use prefixes for large and small numbers

• Table 1-3

• Commonly used prefixes– giga, mega, kilo– centi, milli, micro, nana, pico

Significant Figures

• The leftmost non-zero digit is the most significant figure.

• If there is no decimal point, the rightmost non-zero digit will be the least significant figure.

• If there is a decimal point, the rightmost digit is always the least significant figure.

The number of digits between the Most significant figure and least significant figure inclusive.

Scientific Notation

• Can indicate the number of significant numbers

Significant Figures

• Examples 5 and 6.

• See if you understand them.

Significant Figures

• Multiplication or division.– The least number of significant figures.

• Addition or subtraction.– The smallest number of significant digits on the

right side of the decimal point.

Order of Magnitude

• Table 1-4.

Measurement

• Length– Meter rule– Vernier caliper– Micrometer screw gauge

Practice

Measurement

• Time interval– Stop watch– Ticker tape timer– Timer scaler

Measurement

• Mass– Triple beam balance– Electronic balance

Measurement

• Computer data logging

Error Treatment

• Personal errors– Personal bias

• Random errors– Poor sensitivity of the apparatus

• System errors– Measuring instruments – Techniques

Accuracy and Precision

• Accuracy– How close the measurement to the true value

Precision– Agreement among repeated

measurements– Largest probable error tells the precision

of the measurement

Accuracy and Precision

• Examples 9 and 10

Accuracy and Precision

• Sum and difference– The largest probable error is the sum of the

probable errors of all the quantities.– Example 11

Accuracy and Precision

• Product, quotient and power– Derivatives needed

Kinematics

• Distance d

• Displacement s

Average Velocity

• Average velocity

= displacement time taken

t

svav

Instantaneous Velocity

• Rate of change of displacement in a very short time interval.

dt

sd

t

sv

t

)(lim0

Uniform Velocity

• Average velocity = Instantaneous velocity when the velocity is uniform.

Speed

• Average speed

t

dSpeedav

• Instantaneous speed

t

dSpeed

t

lim0

Speed and Velocity

• Example 13

Relative Velocity

• The velocity of A relative to BBAAB vvv

ABBA vvv

• The velocity of B relative to A

Relative Velocity

• Example 14

Acceleration

• Average acceleration

• Instantaneous acceleration

Average acceleration

• Average acceleration =

change in velocity time

t

vaav

Example 15

Instantaneous acceleration

dt

vd

t

va

t

)(lim0

Example 16

Velocity-time graphv-t graphv

t

Slope: = accelerationdt

dv

v-t graph

• Uniform velocity: slope = 0v

t

v-t graph

• Uniform acceleration: slope = constant

v

t

Falling in viscous liquid

Acceleration

Uniform velocity

Falling in viscous liquid

v

tacceleration:slope=g at t=0

uniform speed:slope = 0

Bouncing ball with energy loss

Falling: with uniform acceleration a = -g.

Let upward vector quantities be positive.

v-t graph of a bouncing ball

• Uniform acceleration: slope = -g

v

t

falling

Bouncing ball with energy loss

Rebound: with large acceleration a.

Let upward vector quantities be positive.

v-t graph of a bouncing ball

• Large acceleration on rebound

v

t

falling

rebound

Bouncing ball with energy loss

Rising: with uniform acceleration a = -g.

Let upward vector quantities be positive.

v-t graph of a bouncing ball

• Uniform acceleration: slope = -g

v

t

falling

reboundrising

v-t graph of a bouncing ball• falling and rising have the same acceleration:

slope = -g

v

t

falling

reboundrising

The speed is less after rebound

Linear Motion: Motion along a straight line

• Uniformly accelerated motion: a = constant

velocity

time

v

u

t0

v

Uniformly accelerated motion

• u = initial velocity (velocity at time = 0).

• v = final velocity (velocity at time = t).

• a = acceleration

t

uv

t

va

v = u + at

Uniformly accelerated motion

• = average velocityv )(2

1vu

time

v

u

t0

v

velocity

Uniformly accelerated motion

time

v

u

t0

v

velocity

s = displacement = tvutv )(2

1

s = area below the graph

Equations of uniformly accelerated motion

tvus

asuv

atuts

atuv

)(2

1

2

2

1

22

2

Uniformly accelerated motion

• Example 17

Free falling: uniformly accelerated motion

Let downward vector quantities be negative

a = -g

Free falling: uniformly accelerated motion

tvus

gsuv

gtuts

gtuv

)(2

1

2

2

1

22

2

a = -g

Free falling: uniformly accelerated motion

Example 18

Parabolic Motion

• Two dimensional motion under constant acceleration.

• There is acceleration perpendicular to the initial velocity

• Examples:– Projectile motion under gravity.– Electron moves into a uniform electric field.

Monkey and Hunter Experiment

gun

bullet aluminiumfoil

electromagnet

iron ball

Monkey and Hunter Experiment

gun

bullet aluminiumfoil

electromagnet

iron ball

The bullet breaks the aluminium foil.

Monkey and Hunter Experiment

gun

bullet

electromagnet

iron ball

Bullet moves under gravity.Iron ball begins to drop.

Monkey and Hunter Experiment

gun

bullet

electromagnet

Bullet is moving under gravity.Iron ball is dropping under gravity.

Monkey and Hunter Experiment

gun

electromagnet

Monkey and Hunter Experiment

gun

electromagnet

The bullet hits the ball!

Monkey and Hunter Experiment

• The vertical motions of both the bullet and the iron are the same.

• The vertical motion of the bullet is independent of its horizontal motion.

Projectile trajectoryy

x

Projectile trajectoryy

x

Projectile trajectoryy

x

u

u = initial velocity = initial angle of inclination

Projectile trajectoryy

x

u

v = velocity at time t = angle of velocity to the horizontal at time t

v

Horizontal line

Projectile trajectoryy

x

u

xu

yu

= x-component of u = y-component of uyuxu

Projectile trajectoryy

x

u

xu

yu

sin.

cos.

uu

uu

y

x

Projectile trajectory:accelerationsy

x

u

xu

yu

ga

a

y

x

0

Projectile trajectoryy

x

u

v

Horizontal line

xu

yuvertical line

xvyv

= x-component of v = y-component of vxvyv

Projectile trajectory: velocity in horizontal directiony

x

u

v

Horizontal line

xu

yu

xv

cos.uuv xx 0xa

Projectile trajectory:velocity in vertical directiony

x

u

v

yuvertical line

yv

tgutguv

ga

yy

y

.sin..

Horizontal line

Projectile trajectory:displacement

y

x x = x-component of s y = y-component of s

s

s = displacement

Projectile trajectory:horizontal displacement

y

x

s

s = displacement

cos..0 uttuxa xx

Projectile trajectory:vertical displacement

y

x

s

s = displacement

22

2

1sin.

2

1gtutgttuy

ga

y

y

Equation of trajectory:a parabolic path

y

x

s

s = displacement

222

.cos2

tan. xu

gxy

Projectile trajectory:direction of motiony

x

u

v

Horizontal line

xu

yuvertical line

xvyv

Angle represents the direction of motion at time t.

Projectile trajectory:direction of motiony

x

u

v

Horizontal line

xu

yuvertical line

xvyv

cos.

sin.tan

u

gtu

v

v

x

y

Projectile trajectory

• Example 19

Projectile trajectory: maximum height H

y

x

u

H

At H, = 0yv g

uH

2

sin 22

Projectile trajectory: range R

y

x

u

At R, y = 0

R

g

uR

2sin2

Projectile trajectory: maximum range Rmax

y

x

u

Rmax

g

uR

2sin2

is maximum when o902

Projectile trajectory: maximum range Rmax

y

x

u

Rmax

R is maximum when o45

Projectile trajectory: maximum range Rmax

y

x

u

Rmax

g

uR

2

max

Projectile trajectory: time of flight to

y

x

u

At time= to , y = 0

R to

g

uto

sin2

Projectile trajectory: two angles for one range

y

x1 R

2

uu

1= - 2o90