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Question

# The acceleration of a particle starting from rest varies with time according to relation $a=\alpha t+\beta$. What will be the velocity of the particle after time $t$?

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Solution

## Acceleration: The rate of change of velocity is defined as acceleration.It is a vector quantity.Step 1: Given dataThe acceleration of the particle, $a=\alpha t+\beta$Step 2: Formula used:We know, The acceleration of a particle,$a=\frac{dv}{dt}$ Where $dv$ is change in velocity and $dt$ is change in time.Step 3: Calculation$a=\alpha t+\beta \phantom{\rule{0ex}{0ex}}\begin{array}{l}⇒\frac{dv}{dt}=\alpha t+\beta \\ ⇒dv=\left(\alpha t+\beta \right)dt\end{array}$On integrating,$\begin{array}{l}{\int }_{0}^{v}dv={\int }_{0}^{t}\left(\alpha t+\beta \right)dt\\ {\left[v\right]}_{0}^{v}={\left[\frac{\alpha {t}^{2}}{2}+\beta t+C\right]}_{0}^{t}\\ v=\frac{\alpha {t}^{2}}{2}+\beta t+C\end{array}$where $C$ is the constant of integration.Since a particle starts from rest, its initial velocity is zero.Therefore when $t=0$, $v=0$.$v=\frac{\alpha {t}^{2}}{2}+\beta t+C\phantom{\rule{0ex}{0ex}}⇒C=0$Therefore, the velocity of the particle after a time $t$ will be$\begin{array}{l}\frac{\alpha {t}^{2}}{2}+\beta t\end{array}$.

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