A particle is moving along x-axis with its position varying with time as The ratio of its initial acceleration, respectively, is [2024]

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Solution

Q.
A particle is moving along x-axis with its position $(x)$ varying with time $(t)$ as $x = α t^{4} + β t^{2} + γ t + δ .$ The ratio of its initial acceleration, respectively, is [2024]

1 $2 α : δ$

2 $γ : 2 δ$

3 $4 α : β$

4 $γ : 2 β$

Ans.
(4)

Given, position, $(x) = α t^{4} + β t^{2} + γ t + δ$

Velocity, $(v) = \frac{d x}{d t} = 4 α t^{2} + 2 β t + γ$

Initial velocity, $v$ $(a t t = 0) = γ$ ...(i)

Acceleration, $(a) = \frac{d v}{d t} = 12 α t^{2} + 2 β$

Initial acceleration, $a$ $(a t t = 0) = 2 β$ ...(ii)

From equation (i) and (ii)

$\frac{v (0)}{a (0)} = \frac{γ}{2 β} = γ : 2 β$

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