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Physics-

General

Easy

Question

A stone dropped from a building of height $h$ and it reaches after $t$ seconds on earth. From the same building if two stones are thrown (one upwards and other downwards) with the same velocity $u$ and they reach the earth surface after $t subscript 1 end subscript$ and $t subscript 2 end subscript$ seconds respectively, then

$t = t_{1} - t_{2}$
$t = \frac{t_{1} + t_{2}}{2}$
$t = \sqrt{t_{1} t_{2}}$
$t = t_{1}^{2} t_{2}^{2}$

The correct answer is: $t equals square root of t subscript 1 end subscript t subscript 2 end subscript end root$

If a stone is dropped from height $h$ then
$h equals fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent$ (i)
if a stone is thrown upward with velocity $u$ then
$h equals negative u blank t subscript 1 end subscript plus fraction numerator 1 over denominator 2 end fraction g t subscript 1 end subscript superscript 2 end superscript$ (ii)
If a stone is thrown downward with velocity $u$ then
$h equals u t subscript 2 end subscript plus fraction numerator 1 over denominator 2 end fraction g t subscript 2 end subscript superscript 2 end superscript$ (iii)
From (i) (ii) and (iii) we get
$negative u t subscript 1 end subscript plus fraction numerator 1 over denominator 2 end fraction g t subscript 1 end subscript superscript 2 end superscript equals fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent$ (iv)
$u t subscript 2 end subscript plus fraction numerator 1 over denominator 2 end fraction g t subscript 2 end subscript superscript 2 end superscript equals fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent$ (v)
Dividing (iv) and (v) we get
$therefore fraction numerator negative u t subscript 1 end subscript over denominator u t subscript 2 end subscript end fraction equals fraction numerator fraction numerator 1 over denominator 2 end fraction g left parenthesis t to the power of 2 end exponent minus t subscript 1 end subscript superscript 2 end superscript right parenthesis over denominator fraction numerator 1 over denominator 2 end fraction g left parenthesis t to the power of 2 end exponent minus t subscript 2 end subscript superscript 2 end superscript right parenthesis end fraction$
Or $negative fraction numerator t subscript 1 end subscript over denominator t subscript 2 end subscript end fraction equals fraction numerator t to the power of 2 end exponent minus t subscript 1 end subscript superscript 2 end superscript over denominator t to the power of 2 end exponent minus t subscript 2 end subscript superscript 2 end superscript end fraction$
By solving $blank t equals square root of t subscript 1 end subscript t subscript 2 end subscript end root$

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