Dr. Jeff's Blog on the Universe

THE SOLUTION TO Weekly Challenge 7: Spaceship Earth

Posted by DrJeff on September 15th, 2009

Filed under 1.1. Dr. Jeff's Weekly Challenge, 1.2. Solutions to Weekly Challenges, 4. The Earth, 5. Space Science, 5.1. Our Solar System, 5.1.1. The Sun, 5.4. Milky Way Galaxy, 5.6. The Universe

Read Original Challenge HERE.

Photo caption: Computer-generated image of the Milky Way galaxy based on real data.

This post is a Dr. Jeff’s Weekly Challenge. It was requested by (teacher extraordinaire) Jami Lupold and her class in the great city of Houston, Texas, USA.

If you’re a teacher and your class has an idea for a blog post, slip me a note!

Last week I gave you a scare. It happened when I told you that you’re really on a spaceship hurtling through space. I was in the midst of describing all of Earth’s motions—it spins, it orbits the Sun, the Sun orbits the center of our galaxy carrying Earth and the Solar System along for the ride—and that’s when I saw panic on your face. You started to get a bit dizzy, so I turned on the “fasten seat belt sign” in light of all the conceptual turbulence ahead. To keep your mind off all the spinnin’ and revolvin’ I gave you an assignment to calculate Earth’s speed—your speed—due to these three motions. Does this all ring a bell? No? Why don’t you go and re-read the original challenge from last week, so you can refocus.

Good. Now that you’re back. Let’s get to the answers. Did I mention this week’s challenge was in our in-flight magazine in the seat pocket in front of you? By the way, I see you dug your fingernails into your seat, and your fingertips seem a bit blue. (Hope the answers don’t send you into a panic.)

And now the answers—

1. The Effect of Earth’s rotation: say you’re just standin’ there on Earth’s equator, all peaceful-like (a shout-out to my favorite class in Texas), minding your own business. How fast are you moving due to Earth’s rotation?

Hint: the Earth rotates once in 24 hours, and think “circumference of Earth.”

Answer: Just STANDING on the equator you are moving about 1,000 mph (nearly 1,700 km/hr)! You’re moving 1 mile every 3 seconds (1 km every 2 seconds)!

See the “How did I come up with those Answers” section below.

2. The Effect of Earth’s revolution (it’s orbit around the Sun):

Now you’re standing on the North Pole. Why? Well it’s a place where we can forget about the speed you’re carried due to Earth’s rotation. Up here (excuse me for a moment …. brrrrr) Earth’s rotation just gently rotates you once in 24 hours on the spin axis you’re standing on. But you’re still movin’ through space, yes you are … ’cause the entire Earth is zipping along in its orbit around the Sun. Your assignment #2 if you choose to accept it: how fast are you moving due to Earth’s motion around the Sun?

Hint: the Sun is 93,000,000 miles or 149,600,000 km (on average) from Earth, and it takes 1 year to go around once. This time think “circumference = 2 x pi x r”

Answer:

STANDING on spaceship Earth, you are being carried around the Sun at a speed of 19 MILES PER SECOND (30 KM PER SECOND)!! Say “one mississippi” …. there, you just moved 19 miles (30 km)! That’s the same as 67,000 miles/hour (107,000 km/hour)!! (Maybe you should keep your seat belt on.)

See the “How did I come up with those Answers” section below.

3. The effect of Earth’s revolution around the center of the Milky Way galaxy:

Your assignment #3 is to figure out how fast the Sun (carrying the entire Solar System along for the ride) is moving in its orbit around the center of the Milky Way galaxy.

Hint: assume the Sun is 28,000 light years from the center, and completes 1 orbit in 240 million years

Oh, other needed Hint: 1 light year = 5.9 trillion miles or 9.5 trillion km

And yeah: think “circumference = 2 x pi x r” (again)

Answer:

For you to orbit the center of the Milky Way once in even the unfathomable time of 240 million years still requires you to be moving very, exceedingly, incredibly, *unbelievably* fast. Right now you are cruising through the galaxy at 495,000 MILES/HOUR (795,000 km/hr)!! That’s 140 MILES/SECOND (220 km/second)! Count out 17 seconds. Cool. You just moved the length of the continental United States, from New York to San Francisco. REALLY!

What? You’d like to travel the entire diameter of planet Earth? Wait 1 minute (actually 57 seconds). Done.

See the “How did I come up with those Answers” section below.

Look up in the sky! It’s a bird! It’s a plane! It’s superman! … big deal. Just sittin’ there, [insert your name here] is WAY faster than a speeding bullet.

Teachers and Parents:

You don’t need to do the calculation with your class (or child at home) to use this post as a powerful lesson. The speeds described above are remarkable, and are sure to impress. Some points:

1. The first calculation is the easiest because you don’t need scientific notation, and you don’t need to convert units. So you might want to do this one to demonstrate the geometry of a circle (the path traveled), and the relation: speed = distance / time.

2. Show some examples of speed = distance / time, e.g., if you travel 100 miles in a car at 50 mph, how long did it take?

3. Have them guess what speeds might be associated with each motion before you tell them the answer.

4. With the answers above in hand, have them calculate how long it would take to travel distances that are familiar to them, e.g., the distance between two cities. Calculate separately the time it would take due to Earth’s rotation, Earth’s orbit around the Sun, and the Solar System’s orbit around the center of the galaxy.

And now—how did I come up with those answers?

Note that calculating answers 2 and 3 below requires an understanding of scientific notation.

There are two key relationships you need for all three questions. In each case the motion you’re investigating carries you once around a circle in a given period of time:

The first is the relationship between a circle’s diameter (d) or radius (r) and the circle’s circumference (c):

circumference = pi x diameter

circumference = 2 x pi x radius

where pi = 3.14

The second relationship provides your speed if you know the distance you’ve traveled in a given time:

speed = distance / time

1. The Effect of Earth’s rotation

The important point: you travel around the circumference of Earth once in 24 hours.

Earth’s equatorial diameter: 7,926 miles (12,756 km)

Earth’s circumference = pi x diameter: 24,888 miles (40,054 km)

speed = distance / time = circumference / time

English units: speed = 24,888 / 24 = 1,037 miles/hr Or: 0.29 miles/sec

Metric units: speed = 40,054 / 24 = 1,670 km/hr Or: 0.46 km/sec

2. The Effect of Earth’s revolution (it’s orbit around the Sun):

The important point: you travel the path of Earth’s orbit around the Sun once in 1 year. You can assume the orbit is a circle whose radius is 93,000,000 miles (149,600,000 km), which is the average distance between Earth and the Sun.

Circumference of Earth’s orbit around Sun:

= 2 x pi x radius: 584 million miles (939,5 million km)

Time to orbit Sun once:

= 1 year = 365.25 days = 8,766 hours = 3.16 x 10 ⁷ seconds

speed = distance / time = circumference / time

English units: speed = 584 x 10⁶ / 8,766 = 66,600 miles/hr Or: 18.5 miles/sec

Metric units: speed = 939.5 x 10⁶ / 8,766 = 107,200 km/hr Or: 29.8 km/sec

3. The effect of Earth’s revolution around the center of the Milky Way galaxy:

The important point: you travel around the center of the Milky Way galaxy once in 240 million years. You can assume the orbit is a circle whose radius is 28,000 light years (each light year is the distance light travels in a year.)

Let’s convert 28,000 light years to meaningful units of miles and km. The hint said that 1 light year = 5.9 trillion miles or 9.5 trillion km

English units: 28,000 light yrs x (5.9 x 10¹²miles /light yr) = 1.65 x 10¹⁷ miles

Metric units: 28,000 light yrs x (9.5 x 10¹² km / light yr) = 2.66 x 10¹⁷ km

Now let’s calculate the circumference of the orbit around the galactic center:

= 2 x pi x radius:

English units: 2 x pi x (1.65 x 10¹⁷ miles) = 1.04 x 10¹⁸ miles

Metric units: 2 x pi x (2.66 x 10¹⁷ km) = 1.67 x 10¹⁸ km

Time to orbit the center of Milky Way once:

240 x 10⁶ years = 8.8 x 10¹⁰ days = 2.1 x 10¹² hours = 7.6 x 10¹⁵ seconds

speed = distance / time = circumference / time

English units: speed = 1.04 x 10¹⁸/ 2.1 x 10¹² = x 495,000 miles/hr

Or: 140 miles/sec

Metric units: speed = 1.67 x 10¹⁸ / 2.1 x10¹² = 795,000 km/hr

Or: 220 km/sec

Photo Credit: National Geographic Society Image Collection, 1999.

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Permalink | September 15th, 2009

2 Responses to “THE SOLUTION TO Weekly Challenge 7: Spaceship Earth”

Benjamin Brooks Says:
September 15th, 2009 at 7:50 pm
Well… I got #2 right,

It’s a shame my attempts at #1 & #3 were out by orders of magnitude. I shall have to watch out for my Scientific notation/carrying/intra-unit conversions…. 🙂
Ben
DrJeff Says:
September 15th, 2009 at 9:16 pm
BB – your approach was perfect! A couple of easy-to-make miscalculations. Great job, and truly thanks for being bold enough to take the challenge! Next week – How Big is Big?! -Jeff