A car accelerates from rest at a constant rate for some time, aft… - Vidyadip

MCQ

A car accelerates from rest at a constant rate $\alpha $ 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
$\frac{{(\alpha + \beta )\,t}}{{\alpha \beta }}$
✓
$\frac{{\alpha \beta \,t}}{{\alpha + \beta }}$

✓

Answer

Correct option: D.

$\frac{{\alpha \beta \,t}}{{\alpha + \beta }}$

d
(d) 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,

$0 = v - \beta (t - {t_1})$ $⇒$ $0 = \alpha {t_1} - \beta t + \beta {t_1}$

$⇒$ ${t_1} = \frac{\beta }{{\alpha + \beta }}t$

$v = \frac{{\alpha \beta }}{{\alpha + \beta }}t$

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