A) \[\frac{1}{{{s}^{3}}}\]
B) \[-\frac{1}{{{s}^{2}}}\]
C) \[\frac{1}{s}-\frac{{{t}^{2}}}{{{s}^{3}}}\]
D) \[\frac{1}{s}-\frac{1}{{{s}^{2}}}\]
Correct Answer: A
Solution :
We have, \[{{s}^{2}}=1+{{t}^{2}}\] ?(i) On differentiating w.r.t. \[t,\]we get \[2s\frac{ds}{dt}=2t\] \[\Rightarrow \] \[\frac{ds}{dt}=\frac{t}{s}\] ?(i) Again differentiating, we get \[{{\left( \frac{ds}{dt} \right)}^{2}}+s\frac{{{d}^{2}}s}{d{{t}^{2}}}=1\] \[\Rightarrow \] \[s\frac{{{d}^{2}}s}{d{{t}^{2}}}=1-{{\left( \frac{ds}{dt} \right)}^{2}}\] \[\Rightarrow \] \[s\frac{{{d}^{2}}s}{d{{t}^{2}}}=1-{{\left( \frac{t}{s} \right)}^{2}}\] [from (ii)] \[\Rightarrow \] \[\frac{{{d}^{2}}s}{d{{t}^{2}}}=\frac{{{s}^{2}}-{{t}^{2}}}{{{s}^{3}}}=\frac{1}{{{s}^{3}}}\] [from (i)]
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