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A stone thrown vertically upward with initial velocity u reach a hight h before coming down.show that the time taken to go up is same as time taken to go down

Asked by Pardeshivinay7 26th April 2019, 7:26 AM
Answered by Expert
Answer:
We need to use the equations of motion, 
 
 v = u + a×t   .....................(1)
v2 = u2 + 2×a×S .................(2) 
 
where v is final velocity after time t, u is initial velocity, a is acceleration and S is distance travelled in time t.
 
 
Let u be the initial velocity of stone.
when the stone reaches the maximum height after time t, its final velocity is zero. 
When the stone travelling vertically upwards, it is experiencing the retardation due to gravity.
 
hence we write the above equation of motion (1) as, 0 = u - g×t or t = u/g .............(3)
 
eqn.(2) will be written as,   0 = u2 - 2×g×h,  or  begin mathsize 12px style u space equals space square root of 2 cross times g cross times h end root space end style...................(4)
 
by substituting u from eqn.(4) in eqn.(3), we have tbegin mathsize 12px style equals space u over g space equals space fraction numerator square root of 2 cross times g cross times h end root over denominator g end fraction space equals space square root of fraction numerator 2 cross times h over denominator g end fraction end root end style............(5)
After reaching maximum height, the stone descends with zero initial velocity, 
accelerated downwards due to gravity and reaches the ground after time t'.
 
Eqns.(1) and (2) are written as follows for downward travel
 
v = 0 + gt'  or  t' = v/g ..........................(6)
 
v2 = 0 + 2gh orbegin mathsize 12px style v space equals space square root of 2 cross times g cross times h end root end style.....................(7)
 
from eqn.(4) and eqn.(7), we can conclude that initial projection velocity and
final velocity when the stone reaches the ground are equal
 
by substituting for v from eqn.(7) in eqn.(6), we get , begin mathsize 12px style t apostrophe space equals space v over g space equals space fraction numerator square root of 2 cross times g cross times h end root over denominator g end fraction space equals space square root of fraction numerator 2 cross times h over denominator g end fraction end root end style................(8)
from eqn.(5) and eqn.(8),  time of ascending the height h is equal to time of descending from same height h.
Answered by Expert 26th April 2019, 1:06 PM
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