Question

A car accelerates from rest at a constant rate α 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. $\frac{(\alpha +\beta )t}{\alpha \beta }$
• D. $\frac{\alpha \beta t}{\alpha +\beta }$

Answer 1

The correct option is D $\frac{\alpha \beta t}{\alpha +\beta }$

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,

$0=v-\beta (t-t_1)\Rightarrow 0=\alpha t_1-\beta t+\beta t_1$
$\Rightarrow t_1=\frac{\beta }{\alpha +\beta }t$
$\therefore v=\frac{\alpha \beta }{\alpha +\beta }t$