question_answer

A car starts from rest and travels with uniform acceleration\[\alpha \]for some time and then with uniform retardation\[\beta \]and comes to rest. If the total time of travel of the car is t, the maximum velocity attained by it is given by                         

(a) \[\frac{\alpha \beta }{(\alpha +\beta )}t\]

(b) \[\frac{1}{2}\frac{\alpha \beta }{(\alpha +\beta )}{{t}^{2}}\]

(c) \[\frac{\alpha \beta }{(\alpha +\beta )}t\]

(d) \[\frac{1}{2}\frac{\alpha \beta }{(\alpha -\beta )}{{t}^{2}}\]

Answer:

Let max. velocity is v. then\[v=\alpha {{t}_{1}}\] and\[v=\beta {{t}_{2}}\] where \[t={{t}_{1}}+{{t}_{2}}\] \[{{t}_{1}}=\frac{v}{\alpha }\]                     \[{{t}_{2}}=\frac{v}{\beta }\] \[t=\frac{v}{\alpha }+\frac{v}{\beta }=v\left( \frac{1}{\alpha }+\frac{1}{\beta } \right)=v\left( \frac{\alpha +\beta }{\alpha \beta } \right)\] \[v=\left( \frac{\alpha \beta }{\alpha +\beta } \right)t\]