A) only on \[\alpha \] and \[\beta \]
B) only on \[\beta \] and \[\gamma \]
C) only on \[\alpha \] and \[\beta \]
D) only on \[\alpha \]
Correct Answer: B
Solution :
Key Idea: Rate of change of displacement is velocity and rate of change of velocity is acceleration. The given equation for displacement is \[x=\alpha {{t}^{3}}+\beta {{t}^{3}}+\gamma t+\delta \] We know that velocity \[=\frac{dx}{dt}\] \[\therefore \] \[v=\frac{dx}{dt}=3{{t}^{2}}\alpha +2t\beta +\gamma \] Initial velocity when t = 0, is \[{{v}_{0}}=3\alpha \times 0+2\beta \times 0+\gamma =\gamma \] ?(i) Also acceleration \[=\frac{dv}{dt}\] \[\therefore \] \[a=\frac{dv}{dt}=\frac{{{d}^{2}}x}{d{{t}^{2}}}=6\,t\,\alpha +2\beta \] Initial acceleration when t = 0, is \[{{a}_{o}}=6\,t\times 0+2\beta =2\beta \] ?(ii) Ratio of initial acceleration to initial velocity is \[\frac{{{a}_{o}}}{{{v}_{o}}}=\frac{2\beta }{\gamma }\] which shown that this ratio depends only on\[\beta \]and \[\gamma \].
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