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Motion In Straight Line

Question
CBSEENPH11017227

A particle is projected vertically upwards from ground. At time it is at A at a height h from ground and still going upwards. From A, it takes further t2 time to reach back to ground again. Show that

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Solution
Let,
Velocity with which the particle is projected from the ground be 'u'. 
The particle takes time t1 to reach A and time t2 to reach the ground. 

Thus, total time of flight of particle is t1 + t2.
Also the total time of fight =  fraction numerator 2 straight u over denominator straight g end fraction
On comparing the two, we get
space space space space space space space space straight t subscript 1 plus straight t subscript 2 equals fraction numerator 2 straight u over denominator straight g end fraction
rightwards double arrow space space space space space space space space space straight u space equals space 1 half straight g left parenthesis straight t subscript 1 plus straight t subscript 2 right parenthesis
 
The particle is at a height h from ground at time t1, which is given by

therefore space space space space space space space straight h space equals space ut subscript 1 minus 1 half gt subscript 1 squared
space space space space space space space space space space space space space space space equals space 1 half straight g left parenthesis straight t subscript 1 plus straight t subscript 2 right parenthesis straight t subscript 1 minus 1 half gt subscript 1 squared
space space space space space space space space space space space space space space space equals 1 half gt subscript 1 straight t subscript 2
(ii) Maximum height attained is,

straight H space equals space fraction numerator straight u squared over denominator 2 straight g end fraction equals fraction numerator open square brackets begin display style 1 half straight g left parenthesis straight t subscript 1 plus straight t subscript 2 end style close square brackets over denominator 2 straight g end fraction

space space space space equals space 1 over 8 straight g left parenthesis straight t subscript 1 plus straight t subscript 2 right parenthesis squared