Help with Question C: Where to Start?

[cloud360]

Homework Statement

I want to know how to answer question C
[PLAIN]http://img546.imageshack.us/img546/5682/maximumvaluek.gif

Homework Equations

The Attempt at a Solution

can somone tell me where to start a least so i can make an attempted solution. i have no idea where to start !

 

[Mark44]

1) v² has to be >= 0, being the square of a real number.
2) u² < 2MG/a, which means that u² - 2MG/a < 0.

In order for the right side to be >= 0, 2MG/r has to be more positive than u² - 2MG/a is negative. IOW, 2MG/r must be > -(u² - 2MG/a).

Does that get you started?

 

[cloud360]

1) v² has to be >= 0, being the square of a real number.
2) u² < 2MG/a, which means that u² - 2MG/a < 0.

In order for the right side to be >= 0, 2MG/r has to be more positive than u² - 2MG/a is negative. IOW, 2MG/r must be > -(u² - 2MG/a).

Does that get you started?

thanks for your reply, i am very grateful :)

but i think of v² as (r')², don't i have to solve a differential? v is not a constant?!? neither is r

NOTE: r'=r dot

 

[Mark44]

Part c is what you asked about, and that doesn't involve solving a differential equation. Since you asked about part c, I assumed you had already done parts a and b.

For what I said earlier, it doesn't matter whether v or r are constants or not.

 

[cloud360]

Part c is what you asked about, and that doesn't involve solving a differential equation. Since you asked about part c, I assumed you had already done parts a and b.

For what I said earlier, it doesn't matter whether v or r are constants or not.

because the thing am confused about is that v^2=r'^2, and i was thinking i need to get r on one side, if so i need to solve the ODE?

if not, i am even more confused as my assumptions were dead wrong.

Would i get the correct answer if i assumed u^2=2MG/a then made the bracket =0, then solved to find r?

i still can't start this :(

hope this is not to much to ask of you, but can you tell me what i am supposed to aim for in this question, do i have to get r on its own on one side?

 

[Mark44]

because the thing am confused about is that v^2=r'^2

They're not asking you to solve this differential equation. Did you do part b? If so, you should have ended up with the equation they show in part b.

, and i was thinking i need to get r on one side, if so i need to solve the ODE?

if not, i am even more confused as my assumptions were dead wrong.

Would i get the correct answer if i assumed u^2=2MG/a then made the bracket =0, then solved to find r?

No, you can't assume that u² = 2MG/a. They already told you that u² < 2MG/a.

I don't know what "made the bracket = 0" means.

i still can't start this :(

hope this is not to much to ask of you, but can you tell me what i am supposed to aim for in this question, do i have to get r on its own on one side?

Reread what I said in post #2. I believe that's the direction you should take. I was able to get an upper bound on r from what I did.

 

[cloud360]

1) v² has to be >= 0, being the square of a real number.
2) u² < 2MG/a, which means that u² - 2MG/a < 0.

In order for the right side to be >= 0, 2MG/r has to be more positive than u² - 2MG/a is negative. IOW, 2MG/r must be > -(u² - 2MG/a).

Does that get you started?

so now should i solve

2MG/r > -(u2 - 2MG/a)?

still finding it difficult to start this question :(

i don't know what am supposed to solve or do to get max value of r.

i am certian max values are find by solving r'=0, they will give criticial points which are either min or max, this is how i have been always thought to find max and min

 

[Mark44]

so now should i solve

2MG/r > -(u² - 2MG/a)?

still finding it difficult to start this question :(

Solve this inequality for r.

i don't know what am supposed to solve or do to get max value of r.

See above.

i am certian max values are find by solving r'=0, they will give criticial points which are either min or max, this is how i have been always thought to find max and min

 

[cloud360]

Solve this inequality for r.
See above.

1) ok, i got

r>-2MG/(u² +2MG/a)

but then r'>0

what to do next

2) also why is it 2MG/r > -(u² - 2MG/a).

why is it not 2MG/r > (u² - 2MG/a),isnt that enough to solve r?

i understand that just being greater than a number i.e 2MG/r > (u² - 2MG/a) won't make RHS positive it has to be 2MG/r > -(u - 2MG/a)

but why are you doing this i.e why are you making sure RHS is positive (i know that r'²>0, meaning RHS has to be positive and also >0), but why isn't 2MG/r > (u² - 2MG/a), enough to solve r, its true isn't it?

3) also , why are you creating an inequality liek this while ignorign r'². is it ok to ignore r'²

 

[Mark44]

1) ok, i got

r>-2MG/(u² +2MG/a)

That's incorrect. You have an incorrect sign somewhere, and you are supposed to show that r is less than some value.

but then r'>0

what to do next

2) also why is it 2MG/r > -(u² - 2MG/a).

why is it not 2MG/r > (u² - 2MG/a),isnt that enough to solve r?

It's given that u² - 2MG/a < 0. The quantity on the left side, 2MG/r, is positive, so saying that 2MG/r is larger than some negative number doesn't help you get an upper bound on r.

The quantity -(u² - 2MG/a) is positive. If we know that 2MG/r is larger than some positive number, we can use that information to say something about r.
So again, solve 2MG/r > -(u² - 2MG/a) for r.

i understand that just being greater than a number i.e 2MG/r > (u² - 2MG/a) won't make RHS positive it has to be 2MG/r > -(u - 2MG/a)

but why are you doing this i.e why are you making sure RHS is positive (i know that r'²>0, meaning RHS has to be positive and also >0), but why isn't 2MG/r > (u² - 2MG/a), enough to solve r, its true isn't it?

3) also , why are you creating an inequality liek this while ignorign r'². is it ok to ignore r'²

The answer is supposed to be an inequality, so you have to start with an inequality.

Yes, you can ignore r'. Part c is not about using r' to solve for r.

 

[cloud360]

Ok i got (after multiplying top and bottom of RHS by a)

r<-(2aGM/(au^2-2Gm)

meaning r'<0, since a,u,G,M are constants?

but we don't need to find r' right? the answer is just r<-(2aGM/(au^2-2Gm) ?

(again i am so grateful to you, for your help)

 

[Mark44]

Ok i got (after multiplying top and bottom of RHS by a)

r<-(2aGM/(au^2-2Gm)

meaning r'<0, since a,u,G,M are constants?

but we don't need to find r' right? the answer is just r<-(2aGM/(au^2-2Gm) ?

(again i am so grateful to you, for your help)

A better way to write this is
r < 2aMG/(2MG - u²), but what you have is also correct.

I don't know how you conclude that r' < 0.

And right, this part doesn't ask you to find r by integrating r'.

 

[cloud360]

A better way to write this is
r < 2aMG/(2MG - u²), but what you have is also correct.

I don't know how you conclude that r' < 0.

And right, this part doesn't ask you to find r by integrating r'.

so is the answer just the maximum is r=-(2aGM/(au^2-2Gm)

or do we have the differentate to get r', also are you saying r' is not <0, since if we differentiate RHS we should get 0, as all of RHS is constant?

 

[Mark44]

so is the answer just the maximum is r=-(2aGM/(au^2-2Gm)

Yes, but you should write this as r_max = 2aMG/(2MG - au^2)

or do we have the differentate to get r', also are you saying r' is not <0, since if we differentiate RHS we should get 0, as all of RHS is constant?

Forget about r'. The equation r_max = 2aMG/(2MG - au^2) tells you the maximum possible value of r(t), which is different from what I'm calling r_max. Both sides are constants, so it would be useless to differentiate both sides.

I'm not saying anything about r'. I don't see anything in this problem that asks you to find r'.