A) \[{{a}^{2}}\frac{{{d}^{2}}t}{d{{v}^{2}}}\]
B) \[{{a}^{3}}\frac{{{d}^{2}}t}{d{{v}^{2}}}\]
C) \[-{{a}^{3}}\frac{{{d}^{2}}t}{d{{v}^{2}}}\]
D) none of these
Correct Answer: C
Solution :
Let \[t=f(v),\]then \[\frac{dt}{dv}=f(v)\Rightarrow \frac{dv}{dt}=\frac{1}{f(v)}\] \[\Rightarrow \]\[a=\frac{1}{f(v)}=1\Rightarrow af(v)\frac{dv}{dt}+\frac{da}{dt}f(v)=0\]\[\Rightarrow \] \[{{a}^{2}}f(v)+\frac{da}{dt}\times \frac{1}{a}=0\] \[\Rightarrow \] \[\frac{da}{dt}=-{{a}^{3}}f(v)\] \[\Rightarrow \]\[\frac{da}{dt}=-{{a}^{3}}\frac{{{d}^{2}}t}{d{{v}^{2}}}\]
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