question_answer

A car accelerates from rest at a constant rate \[\alpha \] for some time. after which it decelerates at a constant rate \[\beta \] and comes to rest. If the total time elapsed is t, then the maximum velocity acquired by the car is

A) \[\left( \frac{{{\alpha }^{2}}+{{\beta }^{2}}}{\alpha \beta } \right)t\]   

B)        \[\left( \frac{{{\alpha }^{2}}-{{\beta }^{2}}}{\alpha \beta } \right)t\]

C) \[\frac{\left( \alpha +\beta  \right)t}{\alpha \beta }\]          

D)        \[\frac{\alpha \beta t}{\alpha +\beta }\]

Correct Answer: D

Solution :

In fig, \[A{{A}_{1}}={{v}_{\max .}}=\alpha {{t}_{1}}=\beta {{t}_{2}}\] But \[t={{t}_{1}}+{{t}_{2}}=\frac{{{v}_{\max }}}{\alpha }+\frac{{{v}_{\max }}}{\beta }\] \[={{v}_{\max }}\left( \frac{1}{\alpha }+\frac{1}{\beta } \right)={{v}_{\max }}\left( \frac{\alpha +\beta }{\alpha \beta } \right)\] or \[{{v}_{\max }}=t\left( \frac{\alpha \beta }{\alpha +\beta } \right)\]