# The Acceleration A Of Particle Starting From Rest Varies With Time According To Relation A=ɑt+𝛃. What Will Be The Velocity Of The Particle After Time T?

Sol:

Acceleration a = $$\alpha t + \beta a = \alpha t + \beta$$

Since a particle starts from rest, its initial velocity is zero

At time t = t = 0 and velocity = 0

$$\frac{dv}{dv} = \alpha t + \beta \\\int_{v}^{0}dv = \int_{t}^{0} (\alpha t + \beta) dt\\V\int_{v}^{0} = (\frac{\alpha t^{2}}{2} + \beta t + {C}’)\int_{t}^{0} \\V = \frac{\alpha t^{2}}{2} + \beta t$$

Therefore, the velocity of the particle after a time t will be $$\frac{\alpha t^{2}}{2} + \beta t$$

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