Question

A car accelerates from rest at a constant rate $\alpha$ for sometime, after which it decelerates at a constant rate $\beta$ to come to rest. If $T$ is the total time, then maximum speed of the car during this time is

• A. $v=\frac{T\alpha \beta }{\alpha +\beta }$
• B. $v=\frac{T\alpha \beta }{\alpha -\beta }$
• C. $v=\frac{T{\beta }^2}{{\alpha }^2-{\beta }^2}$
• D. $v=\frac{T{\alpha }^2}{{\alpha }^2+{\beta }^2}$

Answer 1

The correct option is A $v=\frac{T\alpha \beta }{\alpha +\beta }$

We are given that in the time $T$ the car accelerates with a velocity $\alpha$ and then decelerates with a velocity $\beta$.
Let us consider $t_1$ and $t_2$ are the times during acceleration and deceleration, then
$\frac{v}{t_1}=\alpha$ and $\frac{v}{t_2}=\beta$ and
$T=t_1+t_2=v\left(\frac{\alpha +\beta }{\alpha \beta }\right)=T$
Thus, the maximum speed of the car is $v=T\left(\frac{\alpha \beta }{\alpha +\beta }\right)$