Fearless Boy says ''Let's do some different"

Classical Mechanics Level 5

An fearless boy , want to do stunt . He reach on the top of rough fixed riding track which has coefficient of friction $\mu$ and released his body from rest, so that it leaves the Parabolic( $y=x^2$ ) riding track at the origin, and then finally wish to fall in swimming pool, so that he doesn't hurt.

If value of $R$ is such that , boy will Just fall in the swimming pool. If it is expressed as :

${ R=\sqrt { { me }^{ -\cfrac { \pi }{ n } }-\cfrac { p }{ q } } \\ }$

here $m,n,p,q$ are positive integers and with $p,q$ coprime.

Find $m+n+p+q$

Details and assumptions

$\displaystyle{{ \mu =0.5\\ g=10m/{ s }^{ 2 }\\ H=0.75m\quad \\ h=0.5m }}$
e is Euler's constant.

Try more Deepanshu's Mixing of concept .

The answer is 8.

1 solution

Mvs Saketh
Mar 25, 2015

We know from work energy theorem that

$\displaystyle{\frac { m{ v }^{ 2 } }{ 2 } =mgh-\int { fdl } }$

now let us first expression fdl interms of x and v

$\displaystyle{\int { fdl } =\mu \int _{ x=0 }^{ x=1/2 }{ (mgcos(\theta )+\frac { m{ v }^{ 2 } }{ { r }_{ c } } ) } dl}$

now using $tan\theta =y'\\ \\$ $\displaystyle{\\ { r }_{ c }=\frac { \left( 1+{ (y') }^{ 2 } \right) ^{ 3/2 } }{ y'' } \\ dl=\left( 1+{ (y') }^{ 2 } \right) ^{ 1/2 }dx}$

we get

in the original expression

$\displaystyle{\frac { m{ v }^{ 2 } }{ 2 } =mgh-\mu \int _{ x=x }^{ x=1/2 }{ (mgdx+\frac { y''m{ v }^{ 2 } }{ \left( 1+{ (y') }^{ 2 } \right) } ) } dx}$

now

putting $\displaystyle{y'=2x\\ y''=2}$

$\displaystyle{\frac { m{ v }^{ 2 } }{ 2 } =mgh-\mu \int _{ x=x }^{ x=1/2 }{ (mgdx+\frac { 2m{ v }^{ 2 } }{ \left( 1+{ 4(x) }^{ 2 } \right) } ) } dx}$

now, differentiating both sides with respect to x

and using g=10 and cancelling m on both sides

$\displaystyle{vv'=-20x+5+\frac { 2{ v }^{ 2 } }{ \left( 1+{ 4(x) }^{ 2 } \right) } \\ }$

where v' represents differentiation with respect to x which upon solving yields

$\displaystyle{{ \\ \left| v \right| =\sqrt { C{ e }^{ \tan ^{ -1 }{ (2x) } }-20{ x }^{ 2 }-5 } \\ }}$

which can be solved from boundary conditions easily to get answer

i am sure no one will have trouble from here

And yes, this problem has taught me a wonderful method to deal with weird surfaces with friction where the centrtipetal component cannot be neglected, so thanks @Deepanshu Gupta , this is the best mechanics problem i encountered recently

Nice Problem.

Ronak Agarwal - 6 years, 2 months ago

Congrats on 700 Followers :)

A Former Brilliant Member - 6 years, 2 months ago

Thanks, today boards ended finally, I have to prepare myself for ready set go mains.

Ronak Agarwal - 6 years, 2 months ago

@Ronak Agarwal – I heard that you usually come in the Top 10 in FITJEE AITS , so I think it shouldn't be any problem to you , right ?

A Former Brilliant Member - 6 years, 2 months ago

@A Former Brilliant Member – indeed that is amazing , i have seen AITS paper and if i am right, its perhaps harder than JEE , To score rank 1-10 is really an achievement

Mvs Saketh - 6 years, 2 months ago

Saketh just for a side Note :

Try to use \ (\displaystyle{Enter latex code here}\ ) in latex code it will much better , I had edited your solution , you may refer to that .

Don't give space in b/w :: \ ()

Deepanshu Gupta - 6 years, 2 months ago

Can you please tell me what has swimming pool has to do with this problem.

Ronak Agarwal - 6 years, 2 months ago

ethics, we cant let the boy die, even in a problem (all that ass rubbing down the rough parabolic track was no less harsh)

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – I understand there is need of some liquid having properties of absorbing heat from his ass that's been generated by friction. Also liquids don't have definite shape hence some container is needed so the swimming pool ,

Nice thought Deepanshu, you think on all fronts.

Ronak Agarwal - 6 years, 2 months ago

@Mvs Saketh – XD!! Nice logic

Gautam Sharma - 6 years, 2 months ago

Thanks for the solution missed the centripetal component.

Gautam Sharma - 6 years, 2 months ago

Again Jatin Yadav had also posted a similar problem previously.

Ronak Agarwal - 6 years, 2 months ago

yes but it was much simpler since it was circular, all it took was some clever substitution

Mvs Saketh - 6 years, 2 months ago

Actually I got little inspiration of this question from this math problem

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – Dude what kind of problems do you make ?? I couldn't solve this on my own , had to take help from my friend to do it !

Really nice question :)

A Former Brilliant Member - 6 years, 2 months ago

@A Former Brilliant Member – Thanks for appreciating ... :) I just love to mixing of concepts , So I usually try to mix more than one concepts in one problem , I know I'am unable to create Realistic problems Like Ronak did ... Ronak's problem are really very intersting and very much realistic ...!

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – Currently I'm not able to solve you "Mixed Concepts" questions with ease, I wonder how I'll fare when you start posting Reality based questions !

A Former Brilliant Member - 6 years, 2 months ago

@A Former Brilliant Member – Don't worry I have not that art .... :) , from Since Joining Brilliant I Love 4 guy's problem's most , including Jatin yadav(most realistic) , Ronak , Anish , and Milun moghe (most toughest)

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – No wonder I hadn't heard about Milun Moghe . He hasn't been active for the past 9 Months !

But thanks , I'll try his questions out :)

Btw , I'm a fan of Jatin Yadav's solutions. His methods are innovative and superb. Not that yours isn't !

A Former Brilliant Member - 6 years, 2 months ago

@A Former Brilliant Member – Yes bro try them , He is too-too good in problem making , Even when he is active in his JEE day's then Brilliants call him in California trip .... Really his problems are really very tough !

And Sorry I didn't get you , what did you mean by " Not that yours isn't " ?

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – I meant that even your problem solving methods are innovative :)

A Former Brilliant Member - 6 years, 2 months ago

@A Former Brilliant Member – are aap jaise saare genius 'es ke sammne hum kya chhez hai ....... :) :) btw thanks buddy ! :)

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – Kya baat kar rahe ho bhaiya , maine bhi aap se bahut kuch seekha hai , are umeed karta hoon ki agey bhi seekhtha rahoonga :)

A Former Brilliant Member - 6 years, 2 months ago

@A Former Brilliant Member – are bro ,.... call me as deepanshu... plz .. bhaiya sound's odd... :) , I thought you and me are good friend's , don't you think so ?

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – I'm glad to hear that you think of me as a good friend :) Even I think of you as a good friend and a great inspiration .

Yeah , sorry about that .I'll call you Deepanshu only and nothing else !

A Former Brilliant Member - 6 years, 2 months ago

@A Former Brilliant Member – thanks buddy :)

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – Anytime :)

Well , I gotta go now . You see I have a test paper to write !

A Former Brilliant Member - 6 years, 2 months ago

@A Former Brilliant Member – okay bye , best of luck !

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – No no ! It's not an official test paper . My father asked me to solve some question papers from Reso .

Plus , just as you commented ,I have been engaged by someone else at the moment . I am in the process of replying to him !

A Former Brilliant Member - 6 years, 2 months ago

@A Former Brilliant Member – Yes ... I understand Even Me too going to give paper , of old JEE_MAINS on-line papers , My time is starts at 1.30 PM ... and so I have also got to go ..

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – @Deepanshu Gupta - There have been only two JEE mains, so are you solving the same ones again?

also if you dont mind , what are you doing as your last minute preparation for JEE main?

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – No no ... Every year CBSE conduct's 5 different papers of JEE_mains 1 offline and 4 different online ( computer based ) which are conducting on different dates.
Like in 2014 ::

1)- 6/04/2014- offline paper

2)- 9/04/2014- offline paper

3)-11/04/2014- offline paper

4)-19/04/2014- offline paper

5)-21/04/2014- offline paper

Similarly for JEE_MAINS 2013 And I'am giving all papers , and Yes I gave 2014 paper also , Since I want to compare my mistakes of last year and this year.

If you want link of that papers , then It is here :

JEE_MAINS 2014 all online papers

JEE_MAINS 2013 all online papers

Here you have to do little effort in finding 2013 papers , It will open after following 2-3 Links. But for 2014 , links are direct.

And I don't want do prepartion after 2nd aprail . And I'am only studying Chemistry theory NCERT from now for chapters like , Biomolecules , polymers ,chem. in evryday life and will once again read my weakest topics like Ionic eqm. , Real gas , Surface chemistry , And communications & magnetic properties in physics , and Stastics , and Logic reasoning in maths . And also I continue to give papers with proper time which includes( OMR filling also ) ! After 2nd aprial I will only Learn important formulas , Since It's game of time and , learning them is neccesserry (must) for Mains , that's all I'am thinking to do . what about you ? @Mvs Saketh

Also Please reply to my query in your question here

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – Oh, i recently discovered that and i saw that JEE main online papers in 2013 had some hard problems and some out of course (both JEE and CBSE ) problems for chemistry right?

I found them a bit hard, what about you, alsi do you agree that in 2014 the JEE main online paper of 19th april had too easy chem and math but a slightly harder physics?

I want to know so that i can compare my level of preparation with yours

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – I suggested him to give same paper and compare our level of preparations, I gave 9 April online paper, do you gave it.

Ronak Agarwal - 6 years, 2 months ago

@Ronak Agarwal – how much you get Ronak ?

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – Mine was not good either

Physics-105,Chemistry-95,Maths-120

Ronak Agarwal - 6 years, 2 months ago

@Ronak Agarwal – bro but still you had managed to get 320+ continuously .. I'am sure you will get more on 4th aprail day ...!

have you also filled OMR also in 3 .00 hours ?

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – Deepanshu I always complete paper nearly 35 minutes early , no need of OMR filling

Ronak Agarwal - 6 years, 2 months ago

@Ronak Agarwal – Hey @Ronak and @deepanshu,

Two soap bubbles coalesse to form a single soap bubble, which of the following is true

4PV+3ST=0 3PV+4ST=0 2PV+3ST=0 3PV+2ST=0

V is change in total volume and S is change in surface area,

I suppose you saw it , it is 12 th april problem, could you solve it?

how?

also , did you see the chemsistry of 12 th april, did you find it easy?

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – This is interesting question Seriuosly such question's in Mains ...?? in which paper does it ask ? ... I have only given 9/04/2014 paper yet ...!

I assumed both are identical , then I think Answer should be $displaystyle{3PV+4ST=0}$

my work: Let ${ n }_{ 1 }\& { n }_{ 2 }$ moles in two bubbles after mixing I Think number of moles and Temprature remains constant , Since due to surface tension effect I guess we can't conserve energy .. !

$\displaystyle{{ n }_{ 1 }+{ n }_{ 2 }={ n }_{ f }\\ { P }_{ 1 }{ V }_{ 1 }+{ P }_{ 2 }{ V }_{ 2 }={ P' }{ V }'\\ (P+\cfrac { 4T }{ { r }_{ 1 } } )\cfrac { 4 }{ 3 } \pi { { r }_{ 1 } }^{ 3 }+(P+\cfrac { 4T }{ { r }_{ 2 } } )\cfrac { 4 }{ 3 } \pi { { r }_{ 2 } }^{ 3 }=(P+\cfrac { 4T }{ R } )\cfrac { 4 }{ 3 } \pi R^{ 3 }\quad .\quad .\quad .(1)\\ V=\cfrac { 4 }{ 3 } \pi (R^{ 3 }-{ { r }_{ 1 } }^{ 3 }-{ { r }_{ 2 } }^{ 3 })\quad .\quad .\quad .(2)\\ S=4\pi (R^{ 2 }-{ { r }_{ 1 } }^{ 2 }-{ { r }_{ 2 } }^{ 2 })\quad .\quad .\quad .(3)\\ (1),(2),(3)\\ \\ 3PV+4ST=0}$

PS: I'am not sure of my calculations ....!

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – oh, so you assumed the process is isothermal and got to answer, i thought of that but didnt proceed because nothing was mentioned and when nothing is mentioned, once should assume that energy is conserved (so i assumed that work done by air in bubbles = change in surface energy which gave wrong answer) I think you are right , we assume temperature is constant even in the mercury column questions so same here too, it is in your 11th april paper

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – By the way how much are you getting in 11th April one and 12th April one.

@Mvs Saketh

Also I know how to deal with such problems as soon as I saw it I knew what to do : I left it

Also are you on some other social networking site or maybe we can contact using email and we can discuss further things there, or maybe I post a note on brilliant made solely for discussion.

Ronak Agarwal - 6 years, 2 months ago

@Ronak Agarwal – not very good Phy- 103 ( 3 left ,1 wrong, but i strongly believe that the one question which i showed is perhaps wrong, and another one regarding wavelength, i think answer is incorrect (regarding diffraction)) Chem- 91

maths-120

i have seen that they are making maths too easy in online and physics too wrong and rote, though some questions are pretty interesting as compared to offline version, chem varies from very easy (9th april) to hard as in here

sure, you may mail at &*%^ or make a post, your choice

in 19th april one, i only gave physics so far and got 116 (one question on sky wave frequencis left, but i think physics was really awesome in this paper, just 1 rote problem)

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – Mine I got Physics-102 , Chemistry-112,Maths-115, are you regularly checking emails.

Ronak Agarwal - 6 years, 2 months ago

@Ronak Agarwal – i will from now

Mvs Saketh - 6 years, 2 months ago

@Ronak Agarwal – Not good ? This is not good ? Seriously dude , now you're putting me in a complex !

A Former Brilliant Member - 6 years, 2 months ago

@Ronak Agarwal – @deepanshu- no bro, i have much to learn, lets hope the best for all of use in the exam

Yes i did give, and i found that there were few Rote questions in physics, and i see that such questions are popping everywhere in online papers (like what produces what)

my scores are

Physics - 99 ( lost out on transistor problem, couldnt understand the the problem of one block on another, Didnt know order of Modulli and finally i unfortunately forgot to substitute sign in that galilean telescope when and one more stupid mistake)

Chem- 115 (too easy, just one hard rote learn problem which i guessed incorrectly)

Math - 120

i would add that i used as many Rough sheets as needed and only ticked, not filled OMR

also, i dont think this is so much of reflection of actual performance, the thing is that they make physics a bit harder in online test since you get as many rough sheets as needed as compared to offline

@Ronak Agarwal - Could you solve the one block over another problem?

how?

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – I guess that was given as bonus.

Ronak Agarwal - 6 years, 2 months ago

@Ronak Agarwal – No No Ronak , not any single question of either offline or all online papers are given Bonous ..... in 2014 I have official answer key !

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – You sure @Deepanshu Gupta - what if it was edited out later?

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – Bro... It was CBSE who are now authors of Mains ... they are cruel ... they did not Bonous it .... you can check myCBSEguide.com link i gave above ....

If Question are wrong then gave $-4$ marks to evryone ... Even don't think that it is question of someone life .... It is annoying

Even I suffered it on last year , in bubble Problem in offline paper !

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – Wow ! You guys are really going to be some tough competition for the rest of us !

I could manage only 298 and 300 in two of last years Online Papers .

A Former Brilliant Member - 6 years, 2 months ago

@Mvs Saketh – oh You nailed 334 marks . good to hear it bro ..+

And ohh I see you Get extra 7-8 minutes of OMR filling :)
I have given It by proper method , Like I don't use rough sheet's and Also filling OMR .... and do you know what .... I Enter 1 physics question wrong filled in omr ... So It's also thread for me to fill omr correctly !
I'am really feel bad for physics marks , I have to study Last two NCERT chapter's , Since I directly Leave 3 questions of 12 marks (2 of semiconductor's and 1 of communications ) ... It's really scared me !

And Yes I too have problem in block over block problem .... I guess that was Q-4) am i right ?

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – yes it was, i dont like these guys giving rote learn problems in physics, yes, thats what i said, its quite different when you have to fill OMR as well, not just time, but also the errors that one can do, and yes, the lack or Rough sheets is very crucial, one must be very judicious with paper

And yes, i too faced similar fate regarding the transistor problems(except, the communications one was easy, just look into the communications chapter space waves part) but usually such rote problems are not there in the actual JEE main offline

Hopefully, we will do as well in actual Jee main, i just want above 300

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – Well I guess , You can easily scored 300+ , I also just want to above 300+, I'am really scareing , Since I have to revise (sorry study , Since I will read them after a gap of one year , and also I'am unable to study them with interest) whole NCERT type topics , Like Semi-cond. , commu. , error's analysis (Since I always bunk my Practicals , so don't know about instrument as much ) and too many in chemistry , like surface , chem. in evrday life , polymers (most scaring) , biomolecules and my weakest topic (Real gas)....

what about you , have you complete all such topics ? also what r ur next 2-3 day's targets ?

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – No , Not easy for me, not at all, possible however :)

yes, i can understand, i too find it hard to focus on the boring portions a bit, though i find chemistry pretty easy nowadays, i am stumped with problems that ask you to identify names of reaction unless they are common, i dont remember the names very well, other than that, i might have some trouble with transistors but what i fear more is silly mistakes though i am improving my accuracy and speed as well

my plan is to solve the online papers 2014 and to study all organic reactions from CBSE tommorow, then i shall go through all p-block reactions once again and then the names and formulas of polymers. then i do not know what to do, hence i asked you to get some idea

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – very true " most fear from silly mistakes" I guess Ronak also agree on this , On sunday I gave my Final mock test paper , and I literally Hoping that I had smashed that paper (hoping of ful full in P & M ) ... but what I see , I did that much amount of mistakes , that I had not made in combine 2-3 paper's .

Yes u r right .... I don't know how My marks in chemistry is suddenly increased and physic's decresed exponentially ... ! Hope I will manage to avoid such mistakes on 4th aprail on judgement day

Deepanshu Gupta - 6 years, 2 months ago

@Deepanshu Gupta – Let us hope the best :)

Mvs Saketh - 6 years, 2 months ago

@Mvs Saketh – bro I'am no-where in comparision of your guy's .... and well I have not given all 2014 online papers yet ... Since i have my tution papers also So i'am giving them first ! ... I gave 2014 - 9th aprail online where I got 110 in chem and 120 in maths and extremly low in physics , it was just 83 , I had done so many mistake in that .... how much you got ? ,,,,,, definately it was around 330+ am i right ? In my views 2013 papers of chem. was not as tough ... Although I have to give 3 more Online papers of 2014 yet .. So I can't say right now , about difficullties !

Deepanshu Gupta - 6 years, 2 months ago

I directly used differential form of work energy theorom.

Ronak Agarwal - 6 years, 2 months ago

@Ronak Agarwal – Can you please post a solution of your approach?Please

Gautam Sharma - 6 years, 2 months ago

@Gautam Sharma – My approach is essentially the same as Mvs Saketh's.

Ronak Agarwal - 6 years, 2 months ago

How did you get $dl$ ? @Mvs Saketh

Gautam Sharma - 6 years, 2 months ago

pythagoras theorem, take a small length of curve, it is given by

root((dx)^(2)+(dy)^(2)) = root(1+(y')^(2)) dx (by taking ${dx}^{2}$ common)

now integrate it to find whole length of curve

Mvs Saketh - 6 years, 2 months ago

oh thanks! I was doing that in vectors taking $dxi +dy j$ it was just same but i missed to relate.Thanks again

Gautam Sharma - 6 years, 2 months ago

How is rc formula obtained?

Athul Nambolan - 6 years, 2 months ago

Consider an curve y=f(x) , and again consider an arc element $ds$ at position $x,y$ on it and use pythogorus theoram :

$\displaystyle{ds=\sqrt { { dx }^{ 2 }+{ dy }^{ 2 } } \quad .\quad .\quad .(1)\\ Rd\theta =ds\quad .\quad .\quad .(2)\\ \cfrac { dy }{ dx } =\tan { \theta } \quad .\quad .\quad .(3)\\ ds=dx\sqrt { 1+{ (\cfrac { dy }{ dx } ) }^{ 2 } } =\sec { \theta } dx\quad .\quad .\quad .(4)\quad \quad (from\quad 1\quad \& \quad 3)\\ R=\sec { \theta } (\cfrac { dx }{ d\theta } )\quad .\quad .\quad .\quad .(5)\quad (from\quad 2\quad \& \quad 4)\\ diff\quad eq\quad (3)\\ \cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } =\sec ^{ 2 }{ \theta } \cfrac { d\theta }{ dx } \quad .\quad .\quad .\quad .(6)\\ Use\quad (6)\quad \& \quad (5)\\ \boxed { R=\cfrac { { (1+{ (\cfrac { dy }{ dx } ) }^{ 2 }) }^{ { 3 }/{ 2 } } }{ \cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } } } }$

@Athul Arun A

Deepanshu Gupta - 6 years, 2 months ago

Nice q on mix of mechanics and calculus

aryan goyat - 4 years, 7 months ago

Fearless Boy says ''Let's do some different"

Try more Deepanshu's Mixing of concept .

The answer is 8.

1 solution

0 pending reports