A) \[{{h}_{1}}:{{h}_{2}}\]
B) \[\sqrt{{{h}_{1}}}:\sqrt{{{h}_{2}}}\]
C) \[h_{1}^{2}:h_{2}^{2}\]
D) \[{{h}_{2}}:{{h}_{1}}\]
Correct Answer: A
Solution :
Key Idea: Time taken by the body to fall through the height \[{{h}_{1}}\] is \[{{t}_{1}}=\sqrt{\frac{2{{h}_{1}}}{g}}\] Let \[{{t}_{1}}\] and \[{{t}_{2}}\] be the time taken by the two bodies to fall through the heights \[{{h}_{1}}\] and \[{{h}_{2}}\] respectively, then \[{{h}_{1}}=\frac{1}{2}gt_{1}^{2}\]and\[{{h}_{2}}=\frac{1}{2}gt_{2}^{2}\] \[\Rightarrow \] \[{{t}_{1}}=\sqrt{\frac{2{{h}_{1}}}{g}},\,\,{{t}_{2}}=\sqrt{\frac{2{{h}_{2}}}{g}}\] \[\therefore \] \[{{t}_{1}}:{{t}_{2}}=\sqrt{\frac{2{{h}_{1}}}{g}}:\sqrt{\frac{2{{h}_{2}}}{g}}\] \[\Rightarrow \] \[{{t}_{1}}:{{t}_{2}}=\sqrt{{{h}_{1}}}:\sqrt{{{h}_{2}}}\]
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