A) \[\left( \frac{{{\alpha }^{2}}+{{\beta }^{2}}}{\alpha \beta } \right)\,t\]
B) \[\left( \frac{{{\alpha }^{2}}-{{\beta }^{2}}}{\alpha \beta } \right)\,t\]
C) \[\frac{(\alpha +\beta )\,t}{\alpha \beta }\]
D) \[\frac{\alpha \beta \,t}{\alpha +\beta }\]
Correct Answer: D
Solution :
Let the car accelerate at rate \[\alpha \] for time \[{{t}_{1}}\] then maximum velocity attained, \[v=0+\alpha {{t}_{1}}=\alpha {{t}_{1}}\] Now, the car decelerates at a rate \[\beta \] for time \[(t-{{t}_{1}})\] and finally comes to rest. Then, \[0=v-\beta (t-{{t}_{1}})\]Þ \[0=\alpha {{t}_{1}}-\beta t+\beta {{t}_{1}}\] Þ \[{{t}_{1}}=\frac{\beta }{\alpha +\beta }t\] \ \[v=\frac{\alpha \beta }{\alpha +\beta }t\]
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