question_answer

A car accelerates from rest at a constant rate \[\alpha \]for some time after which it decelerates at a constant rate\[\beta \]to come to rest. If the total time taken is t, find the maximum velocity attained by the car.

Answer:

Let the car accelerate for a time t. Suppose the maximum velocity attained by the car be v. Then\[v=\alpha \,\,{{t}_{1}}\,\,or\,\,{{t}_{1}}=\frac{v}{\alpha }\] Let the car decelerate for a time\[S=ut+\frac{1}{2}a{{t}^{2}}\]. Then\[0=v-\beta \,\,{{t}_{2}}\,\,or\,\,{{t}_{2}}=\frac{v}{\beta }\] Now\[t={{t}_{1}}+{{t}_{2}}=\frac{v}{\alpha }+\frac{v}{\beta }=t\] \[\Rightarrow v=\left( \frac{\alpha \beta }{\alpha +\beta } \right)t\]