Homework help - formula for escape velocity

Nirgal82 · 2002-10-18 20:06:40

Hello all,

I am currently in community college and I was wondering if any of you out there could help me with a pesky math question. (for clarification all numbers that are preceeded by ^ are exponents)

Its the formula for escape velocity where, v = 2Gm/r
(2Gm/r is all under a square root radical sign...don't know how to make them here)
when:

v = escape velocity

m = mass of object (earth = 5.97 * 10^24)

r = radius of object (earth = 6.37 * 10^6)

G = The universal gravitational constant.

This is where it gets tricky for me, because of the way it is printed in my textbook. I have no idea how to read it. I will type it here as it is typed there...

"...G is the universal gravitational constant
which has a value of G = 6.67 * 10 ^-11
m^3/kg * s^2. The mass of Earth is..."

Imagine my confusion. My teacher gave us this as a single question homework assignment. Any help would be appreciated. I don't want you to do the work for me but if you can clarify the universal gravitational constant part, I would be very appreciative...

Your friendly neighborhood, flustered, Martian...
-Matt

Nirgal82 · 2002-10-18 20:17:48

meant to put this in before I sent...

My main confusion is how do I put together the formula for the univerasal gravitational constant, and what m^3/kg * s^2 means...

-Matt

Preston · 2002-10-19 09:38:06

Do you mean you don't know how to read the units? The gravitational constant is

6.67 x 10^-11 (m^3)/(kg*s^3)

so the kilograms and seconds^3 are both in the denominator. Of course, the mass and radius of Earth are in kg and m respectively. Since everything is in the same units, when you calculate the escape velocity, you can ignore all the units and just run it through the calculator... your prof, on the other hand, probably wants the units, but you just treat them like constants in the equation, and they all end up as m/s in the end when you get the velocity.

The units of G are chosed so that the units on both sides of the equation for the force of attraction are the same:

F=(GmM/r^2)

or G=Fr^2/(m*M)

where F is in Newtons, or kg*m/s^2.

Nirgal82 · 2002-10-19 11:41:33

Thats funny, it says s^2 in my book.
And it didn't bother to mention what s meant.
Do you solve for s then? And what value do you enter for kg, 1000?

Thanks a lot Preston,
-Matt

Nirgal82 · 2002-10-19 12:01:18

Actually this question has me very befuddled...
If you or anyone else will go through it step by step with me, I will be very very appreciative...

This is EXACTLY how it appears in my text book...

76.
Escape velocity is the minimum speed
that an object must reach to escape a
planet's pull of gravity. Escape ve-
locity "v" is given in the equation
|----------
v = V 2Gm / r (that is the square root of 2Gm/r)

the planet, r is its radius, and G is the
universal gravitational constant which
has a value of G = 6.67 * 10^-11
m^3 / kg * s^2. The mass of Earth is
5.97 *10^24 kg and its radius is
6.37 *10^6 m. Use this information to
find the escape velocity for Earth.
Round to the nearest whole number.

As you can see, much of my problem lies in how the question was printed in the text. Not to mention the fact that it does nothing to clarify the gravitational constant, namely what value to substitute for "s"

Thanks again...
-Matt

Preston · 2002-10-19 18:42:44

Sorry, that IS seconds^2 (at least I got it right the second time! ) . You don't have to find the the Gravitational Constant, the number is given to you. So all you have to do is plug all the numbers (G, M, and r) into the equation for v. You don't substitute anything for the units kg, m, or s. Those are just units, and as you can see in the equation. The m^3/m cancels to m^2, the kg/kg cancels to 1, and the 1/s^2 stays as it is. That leaves m^2/s^2 under the square root, so when you take the square root, you are left with m/s, or meters/second, which is the units of velocity.

Shaun Barrett · 2002-10-19 19:14:52

Preston has summed it up beautifully!
I could sense major confusion on your part, Nirgal82, and was just trying to figure out the best way to explain it simply when Preston beat me to it. And he did a far better job than I could have, too!

The units of a constant in any self-consistent system of measurement, are automatically designed to largely cancel out when used in a calculation, leaving you with the desired units for the answer. The self-consistency of the system guarantees this - it's automatic.
As Preston explained, you're reading too much into the confusing looking units of the constant, which isn't necessary. If it's any consolation, I've always had a habit myself of looking into mathematical problems too deeply! It caused me no end of trouble in high school. I had to learn to 'go with the flow' and not always be looking for complexity where it doesn't really exist.

But, having said that, units are important. And it's worth chewing over what Preston has told you until you are completely satisfied you understand how SI units work and can comfortably manipulate them. Some people at NASA, working on guidance for a probe called Mars Climate Orbiter, obviously didn't look deeply ENOUGH into their understanding of units!!! I won't dwell on the consequences of that little debacle (too painful) !
???

Phobos · 2002-10-19 19:40:14

But, having said that, units are important. And it's worth chewing over what Preston has told you until you are completely satisfied you understand how SI units work and can comfortably manipulate them. Some people at NASA, working on guidance for a probe called Mars Climate Orbiter, obviously didn't look deeply ENOUGH into their understanding of units!!! I won't dwell on the consequences of that little debacle (too painful) !

Lol, I found it strange that NASA could mix up standard units with metric ones. I always thought NASA worked exclusively with the metric system. I find it funny how so many people in the USA are against the metric system. The metric system is WAY easier to use then the standard one. I can convert instantly distances and volumes from one unit to the next because it's all based on 10 so you usually only need to move decimal points around. The USA system is based on a bunch of abstract numbers with no commonality. 12 inches to a foot, 3 feet to a yard, etc. Good luck trying to instantly convert yards to miles or feet to inches.

Nirgal82 · 2002-10-19 20:28:50

Thanks folks, I havn't the time right now, however I will definately mull over this in the morning...
Thanks Preston, your help is greatly appreciated,
And thanks Shaun, I'm sure you would have been of help if you had made it in time...

-Matt

Shaun Barrett · 2002-10-20 02:08:22

It's incredible to think that, at least up until January 2000, every country in the world except Burma, Liberia, and the U.S.A. had adopted the SI system of measurements.
I could be wrong, but I thought I heard lately that it was down to two countries now. So maybe Burma has changed to metric.
Just in case you're curious, Australia commenced the process of metrication in 1971 and completed the transition by 1981. Changing over is really not as hard as it looks!

Nirgal82 · 2002-10-22 12:40:49

To any who have the knowlege, time, and patience...

Now, nothing major, and I know its a simple piece of vocabulary, but please explain Sidereal Periods to me.
I'm good with conceptualizing most things of this nature, however I still lack a concrete underpinning of this term.

I know it relates to planetary motion, something to do with the amount of time it takes the sun to return to the same place in the sky (or perhaps any other body other than the sun)

If this is the case (or even if it isn't) what would you say about the sidereal period of a planet like say, Mercury?

Thanks again,
Your friendly neighborhood, rapidly educating, Martian...
-Matt

Nirgal82 · 2002-10-22 12:55:50

I must be incredibly dense, because I'm still having a tremendous amount of trouble with the equation for escape velocity...and its due tonight, acck...
Even if it come too late please explain step by step with me.
And if you are the one to do it Preston, please include the equations like you did in your last post, for every step (if you would, I don't know how much of a pain it is to insert those things, so only do it if you want to of course, I don't want to sound bossy)

Thanks,
-Matt

Palomar · 2002-10-22 12:58:02

Just in case you're curious, Australia commenced the process of metrication in 1971 and completed the transition by 1981. Changing over is really not as hard as it looks!

*I remember an 8th-grade math teacher [in 1978] informing the class that "no later than the mid-1990s" the USA would be ENTIRELY converted to the metric system. Well, we're not. Around that time speedometers in vehicles began showing both mph and kph; I never pay attention to the kph, by the way. Our carbonated beverages [soda pop] are sold by the liter...1 or 2 or 3 liter bottles. However, milk, whipping cream, and the like are still sold by quarts, pints, half gallons, and gallons.

It's 2002 and there's no sign of the USA changing totally over to metric anytime soon, unless I'm missing out on hearing about schools pushing it hard on today's generation of school kiddies.

My husband is a mechanic, and of course enjoys building engines, tearing things apart, etc. Some of his tools are standard and others are metric, as are some of the equipment he works on.

What a mess.

I could never understand the Fahrenheit to celsius exchange, though...and it's been a long time since I tried. ???

NASA had better get its act together about this!!

Phobos, as for why many of our fellow Americans are resistant to the switch: I have to admit I've been resistant to it as well. It's dislike of loss of familiarity, I guess. ???

--Cindy

Shaun Barrett · 2002-10-22 19:27:05

Hi Matt!

I think Preston's neatly produced square roots, brackets, and exponentials are probably your best hope of understanding this stuff. I don't seem to be able to duplicate that sort of professional looking presentation!
Also, I'm not quite sure at what level you are having difficulty.

I assume you are comfortable with exponentials? Looking at Preston's nicely laid out equation, you can see that on the top line we have a 10^-11 and a 10^24. Multiplying these together gives us 10^13. On the bottom line, we have just one exponential, 10^6. This means we now have 10^13 divided by 10^6, which gives us 10^7. Keep that in mind.
Now for the numbers!
This is the easy part. On the top line we have 2 x 6.67 x 5.97. This equals 79.64 (to two decimal places). On the bottom line, we have just one number, 6.37. Now all we have to do is divide 79.64 by 6.37, which equals 12.5 (to two decimal places again).

So now we have, inside the square root sign, 12.5 x 10^7. For ease of 'square rooting', let's manipulate that a bit and call it 125 x 10^6.
Now we can 'square root' it to 11.18 x 10^3, or 11,180.
But 11,180 what? In the SI system, we should end up with metres per second for a velocity calculation. Is that what we get if we look at the units? Let's see!

In the same way that the powers-of-ten terms (or exponentials) can be cancelled top and bottom, the way we did above, so too can the units be cancelled top and bottom.
Referring to Preston's lay out again, on the top line we have m^3 and kg. On the bottom line, we have m, kg, and s^2. Cancelling like terms, we are left with m^2 on the top line, and s^2 on the bottom line.
But they, too, are inside the square root sign and need to be 'square rooted'!
Doing that leaves us with m on top and s underneath. Which is the m/s unit we wanted! The self-consistent system of units has done its job!

And the final answer is that Earth's escape velocity is 11,180 m/s ( or roughly 7 miles per second for our American cousins! )

I have no idea whether this has been helpful to you, Matt. I can only hope!
It's probably too late for your assignment deadline, though. If so, I apologise.
Good luck!!

Phobos · 2002-10-22 20:17:08

Phobos, as for why many of our fellow Americans are resistant to the switch: I have to admit I've been resistant to it as well. It's dislike of loss of familiarity, I guess.
--Cindy

Yeah, the loss of familiarity is the big hurdle. I think some Americans though consider switching over to the metric system to be something akin to an act of godless communism. The imperial system of units seems to be more than just a measuring system, it's a point of American pride right up there with mom and apple pie. I don't know how many arguments I've gotten into with people over the pros of converting to the metric system and the arguments usually turn into some weird emotional thing about how the metric system is the work of the devil or something like that. You'd be surprised! ???

Nirgal82 · 2002-10-22 21:49:28

Thanks Shaun, Preston, others...

Wave after glorious wave of understanding have left me euphoric.
I knew it was somewhat simple, I was just blind to it.
I forgot you could simply add exponents when multiplying, also the quotient rule for exponents and radicals, which helped bring the exponent from the bottom to the top.
Once that was done everything else came together, it seems I was intimidated by the scientific notation.
My road to becoming a scientist has barely been made visible to me, and some hurdles I have already past, and I know of a few I will have to tackle later...
For anyone who is interested I am scoring 100s or over in my Geology class (over due to extra credit assigments)
And is it Arizona State University at Phoenix that has the planetary geology programs? Do they have specific requisites that I may have to accomplish in order to be accepted there?

Thanks again for all the help, It was much appreciated.
And as for the fate of this thread, I'm sure I can keep it going with fresh questions such as, "Sidereal period, example: Mercury" <~~~any takers?
I encourage any other students floating around these forums to utilize this, not only for help, but perhaps a little something here and there to broaden the horizons of others...

Your friendly neighborhood, appreciative, Martian...
-Matt Gillespie

Palomar · 2002-10-23 11:34:06

Phobos, as for why many of our fellow Americans are resistant to the switch: I have to admit I've been resistant to it as well. It's dislike of loss of familiarity, I guess.
--Cindy
Yeah, the loss of familiarity is the big hurdle. I think some Americans though consider switching over to the metric system to be something akin to an act of godless communism. The imperial system of units seems to be more than just a measuring system, it's a point of American pride right up there with mom and apple pie. I don't know how many arguments I've gotten into with people over the pros of converting to the metric system and the arguments usually turn into some weird emotional thing about how the metric system is the work of the devil or something like that. You'd be surprised! ???

*I've been thinking your response over, and you might be onto something. The US is a rather religious nation, and at least in the 1980s [and prior to] a lot of Christian sects were opposed to "one-world-ization." I seem to recall that many Christians I grew up with disliked the metric conversion plans because it smacked of some nefarious plan to further advance the cause of a "one-world government."

::shrugs::

--Cindy

Nirgal82 · 2002-10-23 12:03:02

Uhh...
Not to be pushy or anything like that. This is a thread related to homework, not necessarily the merits of metric conversion.

Don't get me wrong, I do think metric is the way to go, and the ridiculous arguments Americans in particular make against metric are a little weird.
They counter with strange fears and stuff like that, its almost as if you asked them to question their faith in their god when all you asked them to do was question their system of measurments. (even though, even for the religious questioning their god would be a good thing, I mean, if your faith is actually worth having faith in, it should be able to withstand questioning...but that my humble opinion and I'm sure that many people would call me a sinner and damn me to hell for it, or worse, the most devious of christian tactics would be employed, they would feel sorry for me and try to "save" me)
Anywho, I'm sure a metric thread will do fine here, I may create one, if someone hasn't beaten me to it, so as to better keep this thread on Topic...

Your friendly neighborhood Martian...
-Matt

Nirgal82 · 2002-11-26 11:08:39

Okay, I'm back, after some downtime...working on school and other ventures...

Heres a question from my AstroLab class that is relevant here...

The question is written as follows:

We can't just return to Earth anytime we'd like to (from Mars). We have to wait to take off (in order to conserve fuel) until the Earth will be in the proper positioin at the end of our transfer orbit home. In general, we will have a long wait, but we'll assume we have the provisions necessary to make it through (and in the meantime, explore Mars)

An accurate calculation of the time we will have to wait is complicated, but we can put an upper limit on it by realizing that after on synodicc period the Earth will be in the same place, relative to Mars, as it was when we landed. Surely the two planets will go through the proper configuration for launch before then. Using this formula:

1/Psyn = (1/Pearth) - (1/Pmars)

Psyn - synodic period of Earth and Mars
Pearth - Earth's orbital period
Pmars - Mars's orbital period

Calculate what the synodic period would be.

This is where I hit my roadblock. Though this formula looks simple, I cannot seem to get the right answer, unless .465240641711 is correct. (which I doubt)

Anyone out there, preston I'm looking at you, heh heh...

Thanks
Your friendly neighborhood busy body Martian...
-Matt

CalTech2010 · 2002-11-26 11:52:07

Phobos said something to the effect of "I can convert units easier with the metric system because it is based on 10."

Because of my Physics classes in High School, I had to get adjusted to using the metric system. My main problem was that I could do conversions very quickly, but I had a hard time applying what I'd converted. With the Imperial system, I could look down a road and say, "that hill is 3 miles off" or, "there must be 5 gallons of water in that bucket" and I could eyeball my measurements. With the metric system, I had to relearn my approximated distances before I could begin to apply the system.

I think many Americans are afraid of the transition to metrics because they would lose their ability to approximate size.

Just a thought... -CALTECH2010

PS If you want to convert to the metric system, run track in High School. It helped me to be able to guage distances using our track size (I still use it to recall how long 100 m is!)

Preston · 2002-11-26 14:09:54

0.469 (year^-1) is the inverse of the synodic period. You have to take the inverse of that to get the synodic period, 2.13 years.

In other words Psyn = (1/Pearth - 1/Pmars)^-1

Phobos · 2002-11-26 21:14:17

With the Imperial system, I could look down a road and say, "that hill is 3 miles off" or, "there must be 5 gallons of water in that bucket" and I could eyeball my measurements. With the metric system, I had to relearn my approximated distances before I could begin to apply the system.
I think many Americans are afraid of the transition to metrics because they would lose their ability to approximate size.

I had the same problems myself especially when it came to weights. I used to always have to remind myself that a kg was more than a pound not less. Grams are such miniscule amounts of weight that you just kind of automatically think that a kg is light to. Yeah that's probably one of the main problems with switching systems, you lose that intuitive ability for awhile.

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#1 2002-10-18 20:06:40

Re: Homework help - formula for escape velocity

#2 2002-10-18 20:17:48

Re: Homework help - formula for escape velocity

#3 2002-10-19 09:38:06

Re: Homework help - formula for escape velocity

#4 2002-10-19 11:41:33

Re: Homework help - formula for escape velocity

#5 2002-10-19 12:01:18

Re: Homework help - formula for escape velocity

#6 2002-10-19 18:42:44

Re: Homework help - formula for escape velocity

#7 2002-10-19 19:14:52

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#8 2002-10-19 19:40:14

Re: Homework help - formula for escape velocity

#9 2002-10-19 20:28:50

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#12 2002-10-22 12:55:50

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#14 2002-10-22 19:27:05

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#16 2002-10-22 21:49:28

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#17 2002-10-23 11:34:06

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Re: Homework help - formula for escape velocity

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#20 2002-11-26 11:52:07

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