First slide

Rectilinear Motion

Question

A car accelerates from rest at a constant rate $α$ for some time after which it decelerates at a constant rate $β$ to come to rest. If the total time elapsed is “t” then the maximum velocity attained is

Difficult

A

$\frac{α β t}{(α + β)}$
B

$\frac{α β t}{2 (α + β)}$
C

$\frac{2 α β t}{(α + β)}$
D

$\frac{4 α β t}{(α + β)}$

Solution

$f r o m A t o B : i n i t i a l v e l o c i t y = 0; f i n a l v e l o c i t y = V; a c c e l e r a t i o n = α; t i m e = t_{1} s u b s t i t u t e i n V = u + a t, ---(1) V = α t_{1} \Rightarrow t_{1} = \frac{V}{α} ---(2) f r o m B t o C : i n i t i a l v e l o c i t y = V; f i n a l v e l o c i t y = 0; a c c e l e r a t i o n = - β; t i m e = t_{2} s u b s t i t u t e i n e q u a t i o n (1) 0 = V - β t_{2} \Rightarrow t_{2} = \frac{V}{β} ---(3) a d d e q u a t i o n s (2) a n d (3)$
$t_{1} + t_{2} = t$
$\frac{V}{α} + \frac{V}{β} = t$
$\Rightarrow V = \frac{α β t}{α + β}$

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