First slide

Instantaneous acceleration

Question

For a particle moving in a straight line, the displacement s depends on time t as $t = {αt}^{3} + {βt}^{2} + γt + δ$ . The ratio of its initial acceleration to its initial velocity is

Moderate

A

$α / 2 β$
B

$γ / 2 β$
C

$2 β / γ$
D

$2 β / α$

Solution

Given that, $s = {αt}^{3} + {βt}^{2} + γt + δ$
Now, $v = \frac{ds}{dt} = 3 {αt}^{2} + 2 βt + γ . . . . . (1)$
and $a = \frac{dv}{dt} = 6 αt + 2 β . . . . (2)$
At time t = 0,
From eq. (1), $v = γ$
From eq. (ii), $a = 2 β$
$∴ {(\frac{a}{V})}_{t = 0} = \frac{2 β}{γ}$

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