An IntroducEon to Problem Solving (Some examples

Quan%ta%ve Ac%vi%es for Introductory Astronomy 1 1,2 1,3 Jonathan W. Keohane , Jennifer L. Bartle4 & Joseph P. Foy 1Hampden-Sydney College, 2USNO, 3Barre4 Honors College at ASU

The Astronomical Scien%fic Method •  Astronomy is not an experimental science. We observe nature, and try to figure it out. •  Usually, farther objects are harder to study than close-by ones. •  We study a few close or bright objects very well, and then begin by assuming that similar objects are idenPcal except for what we can actually measure. •  Thus, we always measure far away objects in comparison to similar well-known objects. •  Stars are measured in raPo to the sun or a bright star like Vega. •  Planets are measured in raPo to either the Earth or Jupiter. •  So, to answer quesPons, we find a proporPonality and use it to compare something new to something well-known.



An Introduc%on to Problem Solving (Some examples from lecture) Example Problem #1 -- Driving with Mom Problem: •  You drive to school with your mom. On the way, you carefully follow the posted speed limit, and it takes you 3h45m to get here. •  Later you drive the same trip by yourself, and it takes you 3h15m to get here. •  By what average percentage were you over the speed limit the second Pme you came to school? Solu%on: Main Point:

Organiza%on Table of Data: We must make sure we have everything in the same unit, between our reference and what we are interested in. The unit does not ma4er, as long as it is the same. Then we fill in a table of what we know. 3h 45m = (3 x 60 + 45) min = 225 min 3h 15m = (3 x 60 + 15) min = 195 min

Because the distance is the same, the speed (v) is inversely proporPonal to the travel Pme (t).

d t d v= t v alone = v mom v=

v alone = v mom

Driving Alone

Driving with Mom

( )= ( ) 1 t alone

1

1

t alone t mom

t mom

1 ⎛ 195 min ⎞ ⎜ ⎟ ⎝ 225 min ⎠

=

225 min = 1.15 195 min

So, you drove 15% over the speed limit – on average.

QuanPty

Driving Alone

Driving with Mom

Units

Velocity

?

1

Mom Units

Time

195

225

Minutes

Distance

1

1

Mom Units

Or about 63 mph in a 55 mph zone.

Example Problem #2 -- The Orbit of Jupiter Problem: •  It takes Jupiter 11.86 years to orbit the sun, with respect to the background stars. How far away is Jupiter from the sun? Express your answer in A.U.

PJupiter rJupiter rEarth

Solu%on:

PEarth

Main Point: Both Jupiter and the Earth orbit the Sun. We use Kepler’s 3rd law and compare Jupiter to the Earth.

M P2 ∝ r 3

Organiza%on Table of Data:

M sun P 2 Jup r 3Jup = M sun P 2 Earth r 3Earth 3

⎛ rJup ⎞ ⎛ PJup ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ r P ⎝ Earth ⎠ ⎝ Earth ⎠ rJup

2

2

⎛ P ⎞ ⎛ P ⎞ = ⎜⎜ Jup ⎟⎟ = ⎜⎜ Jup ⎟⎟ rEarth ⎝ PEarth ⎠ ⎝ PEarth ⎠ rJup = 5.2 AU 3

2

3

⎛ 11.86 ⎞ =⎜ ⎟ ⎝ 1 ⎠

2

3

Quantity

Jupiter

Earth

Unit

Period

11.86

1.00

Years

SemiMajor Axis

?

1.00

AU

Mass

1.00

1.00

Solar Mass

Example Problem #3 -- The Mass of Saturn Problem: •  Saturn’s largest moon, Titan, is located approximately 1,222,000 km from its center, and it takes 15.95 days to complete an orbit? •  Our moon has an orbit that is 384,400 km in radius PTitan and a period of 27.32 days. rPtan •  How may Pmes more massive is Saturn than the Earth?

PMoon rMoon

Saturn

Earth

Solu%on: Main Point: Titan orbits Saturn, so we can find the mass of Saturn by knowing about Titan’s orbit. We use Kepler’s 3rd law and compare Titan to the Moon and Saturn to the Earth.

M P2 ∝ r 3

← Kepler' s 3rd Law

r3 M∝ 2 P

Organiza%on Table of Data:

← Divide both sides by P2

Quantity

Titan / Saturn

Moon / Earth

Unit

Period

15.95

27.32

days

SemiMajor Axis

1,222,000

Mass

?

3

M Saturn M Earth

M Saturn M Earth

⎛ rTitan ⎞ ⎜⎜ ⎟⎟ r = ⎝ Moon ⎠ 2 ⎛ PTitan ⎞ ⎜⎜ ⎟⎟ P ⎝ Moon ⎠

← Solve as a ratio

⎛ 1,222,000km ⎞ ⎜⎜ ⎟⎟ 384 , 400 km ⎠ =⎝ 2 ⎛ 15.95days ⎞ ⎜⎜ ⎟⎟ 27 . 32 days ⎝ ⎠

3

← Note : units cancel

384,000 km 1.00

Earth Masses

3

(3.179) = 94.25 M Saturn = M Earth (0.5838)2

So, Saturn is about 95 Pmes more massive than the Earth.