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Physics-

General

Easy

Question

A body is projected vertically upwards at time $t equals 0$ and it is seen at a height $H$ at instants $t subscript 1 end subscript$ and $t subscript 2 end subscript$ seconds during its flight. The maximum height attained is $left parenthesis g$ is acceleration due to gravity $right parenthesis$

$\frac{g {(t_{2} - t_{1})}^{2}}{8}$
$\frac{g {(t_{1} + t_{2})}^{2}}{4}$
$\frac{g {(t_{1} + t_{2})}^{2}}{8}$
$\frac{g {(t_{2} - t_{1})}^{2}}{4}$

The correct answer is: $fraction numerator g open parentheses t subscript 1 end subscript plus t subscript 2 end subscript close parentheses to the power of 2 end exponent over denominator 8 end fraction$

Let time taken by the body to fall from point $C$ to $B$ be $t ´ ´$ Then, $t subscript 1 end subscript plus 2 t to the power of ´ ´ end exponent equals t subscript 2 end subscript$
$t to the power of ´ ´ end exponent equals open parentheses fraction numerator t subscript 2 end subscript minus t subscript 1 end subscript over denominator 2 end fraction close parentheses$ (i)
Total time taken, to reach point $C$

$T equals t subscript 1 end subscript plus t ´ ´$
$equals t subscript 1 end subscript plus fraction numerator t subscript 2 end subscript minus t subscript 1 end subscript over denominator 2 end fraction$
$equals fraction numerator 2 t subscript 1 end subscript plus t subscript 2 end subscript minus t subscript 1 end subscript over denominator 2 end fraction equals open parentheses fraction numerator t subscript 1 end subscript plus t subscript 2 end subscript over denominator 2 end fraction close parentheses$
Then maximum height attained
$H subscript m a x end subscript equals fraction numerator 1 over denominator 2 end fraction g open parentheses T close parentheses to the power of 2 end exponent equals fraction numerator 1 over denominator 2 end fraction g open parentheses fraction numerator T subscript 1 end subscript plus t subscript 2 end subscript over denominator 2 end fraction close parentheses to the power of 2 end exponent$
$equals fraction numerator 1 over denominator 2 end fraction g. fraction numerator open parentheses t subscript 1 end subscript plus t subscript 2 end subscript close parentheses to the power of 2 end exponent over denominator 4 end fraction$
$rightwards double arrow H subscript m a x end subscript equals fraction numerator 1 over denominator 8 end fraction g. open parentheses t subscript 1 end subscript plus t subscript 2 end subscript close parentheses to the power of 2 end exponent m$

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