Question

A stone is dropped from the top of a cliff of height h . n seconds later a second stone is projected downwards from the same cliff with a vertical velocity u.The condition so that both the stones meet at the bottom of the cliff together is?

Answer 1

x= Ut + 1/2 at2 ..

U = initial velocity,
a= Accelaration = g
t = time required to reach ground..

Initial Energy is Potential energy, U = mgh..
As height and gravity remains same, Only factor changing will be Mass of the stone..

I guess, this can help u !!

Answer 2

this is what i got so far:
if stone 1 takes time t to reach the ground. then stone 2 should take time (t-n) for them both to meet at the bottom.
so I got t = (gn^2 - 2un)/ (2u + 2gn)
if we substitute this in
h = -1/2 gt^2...........we should get a relation. but this seems to be complicated. anyone got better ideas?

Answer 3

Let after n seconds first stone takes t seconds to reach the bottom.
So first stone takes totally (n+t) seconds to reach bottom of the cliff .
And then the second ston
]e takes totally t seconds to reach bottom (as both has to meet at the bottom)

then the motion-eqn for first stone is:
\[h=(1/2)g(t+n)^{2}...................(1)\](as it is simply dropped so no initial velocity,and g is acceleration due to gravity)

and the motion-eqn for the second stone is:
\[h=ut+(1/2)g.t^{2}...................(2)\](as it has initial velocity u)

then adding (1) and (2) and separating h we get:

\[(1/2)g.(t+n)^{2}=ut+(1/2)g.t^{2}\]
\[(1/2)g[(t+n)^{2}-t^{2}]=ut\]
\[(1/2)g[(2t+n).n]=ut\]
\[2tn+n^{2}=2ut/g\]
\[2t[(u/g)-n]=n^{2}\]
\[t=(1/2)[n^{2}g/(u-ng)]\]

Answer 4

@ganpat how can mass of the stone change? if we take that approach then we'll have to consider KE at the bottom. so mgh = 1/2 mv^2

Answer 5

@Vaidehi09 : The Second stone can be of different mass..

This can be a parameter for them to meet ??

Answer 6

@Vaidehi09 mass has nothing to do here in kinematics. I hope you know that all particle or object irrespective of their mass fall in same time from same height.

Answer 7

but since mass is not mentioned in the prob. we can't have a relation in terms of it.

Answer 8

@biswajit_paul yes, i'm aware.

Answer 9

if not mass than, Area can be the factor... Surface are ??

Answer 10

ok @Vaidehi09 so their is no problem if the masses of the stones are different. And obviously we have neglect air friction or other dssipative force .

Answer 11

@Ganpat the surface area has nothing to do here as we have not considered friction.

Answer 12

we need to find a relation in terms of h, u, n, g...most probably. coz these terms are specified in the question. so any extra terms we take while solving, need to be eliminated. so why would we add extra terms as mass and area here?

Answer 13

hello..I have said you that mass is here useless term. Forget about it..

Answer 14

i know. forget mass and area.

Answer 15

umm...so what abt the answer here? @DLS what is it?

Answer 16

@Vaidehi09 havent you seen my reply? I have given an answer with derivation. Whats about it?

Answer 17

don't we have to eliminate t? since t is a variable chosen by us. it isn't given to us in the question.

Answer 18

v(6n-9)/2n

Answer 19

What I have found is also a condition..What do you think?

Answer 20

@mukushla

Answer 21

@DLS how did you find out the condition you have given here?

Answer 22

That is the answer

Answer 23

ok. let me try

Answer 24

@DLS I hope that the v will be u. Isn't it?

Answer 25

@biswajit_paul yes, that is a condition. I also got something along the same lines. but during substitution it got complicated.

Answer 26

yes...that v should be u.

Answer 27

be back in 15 mins!

Answer 28

@DLS this is condition for what? h?

Answer 29

hello @DLS

Answer 30

@DLS no...that v should be u. or check the ques again. what is given in the ques - v or u? also...the answer...is it in the form of an equation or its just an expression?

Answer 31

It is v only
and answer is what i posted earlier only

Answer 32

@mukushla @Vaidehi09 I'n not getting 2 in the denominator :/

Answer 33

how did u proceed?

Answer 34

i did from the beginning

Answer 35

See,distance travelled in n seconds =>
sn = ut+1/2at^2
And distance travelled in nth second is given as:
Sn = S(t-1) = u+a/2(2t-1)
I usd this equation to get something good :o try it,maybe helps..

Answer 36

@mukushla

Answer 37

how come the answer is not in the form of an equation?

Answer 38

i dont know if this is right or not
\[n=\frac{v+\sqrt{2gh}-\sqrt{v^2+2gh}}{g}\]

Answer 39

i got this eq for time t:
\[t = gn ^{2} \div 2( v - gn )\]

Answer 40

but the final ans has doesn't have g, t, or h.

Answer 41

what is the final answer?

Answer 42

v(6n - 9) / 2n
what's more....its not even an equation!

Answer 43

!!!!

Answer 44

i know, right!

Answer 45

Like I said,distance travelled in nth second is given as:
Sn = S(t-1) = u+a/2(2t-1)
use this and get the answer !!

Answer 46

how did u get that eq?

Answer 47

I got it in "Terms" of that :o im not substituting values right i guess :/

Answer 48

u mean u derived it? how? show it here.

Answer 49

I just have that as a formula :P

Answer 50

i can't recognize it.....

Answer 51

private formulas~

Answer 52

arre just tell the time's value..will it be n-3?

Answer 53

t is the time taken for stone1 to reach the ground ryt? so stone 2 will take t-n time.

Answer 54

OH WTH SORRY THIS IS SOME OTHER QUES LOL SORRY :/

Answer 55

hi friends
@DLS could u plz check the final answer again ?

Answer 56

ANSWER IS !
8h(u-gn)^2 = gn^2(2u-gn)^2
sorry,i thought this was the next ques which i ws doing!

Answer 57

@DLS ...................................................seriously, u had us racking our brains for sooooo long when we already had the answer.

Answer 58

lol..sorry :/ i thought i posted the ques i was doing :/ im sorry..

Answer 59

thats what i thought n=f(u,h)

Answer 60

oh well....its fine already. so we solved this one too.

Answer 61

arre solution :P ?

Answer 62

ok...proceed this way:
h = 0 + 0.5gt^2............stone 1..............=> (i)
h = u(t-n) + 0.5g (t-n)^2............stone 2
then equate these 2....coz h is on LHS of both.
get the equation for t
substitute that in (i)
u'll get the answer.

Answer 63

hm, I have also got this answer.

Answer 64

@biswajit_paul hi5!

Answer 65

@Vaidehi09 yes