A particle is moving along the path given by $y = \frac{C t^{6}}{6}$ where $C$ is a positive constant. The relation between acceleration $(a)$ and the velocity of the particle at $t = 5 s$ is given by $a = η v$ , then the value of $5 η$ is

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Solution

Given; $y = \frac{C t^{6}}{6}$
Differentiating once w.r.t. time,
Velocity $= v = \frac{d y}{d t} = C t^{5}$
Differentiating once more w.r.t. time,
Acceleration $= a = \frac{d v}{d t} = 5 C t^{4} \Rightarrow \frac{a}{v} = \frac{5}{t} = η$
$\Rightarrow η = 1 (∵ t = 5)$
$\Rightarrow 5 η = 5$

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A particle is moving along the path given by $y = \frac{C t^{6}}{6}$ where $C$ is a positive constant. The relation between acceleration $(a)$ and the velocity of the particle at $t = 5 s$ is given by $a = η v$ , then the value of $5 η$ is

Given; $y = \frac{C t^{6}}{6}$
Differentiating once w.r.t. time,
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Differentiating once more w.r.t. time,
Acceleration $= a = \frac{d v}{d t} = 5 C t^{4} \Rightarrow \frac{a}{v} = \frac{5}{t} = η$
$\Rightarrow η = 1 (∵ t = 5)$
$\Rightarrow 5 η = 5$