Question

the initial velocity of a particle is u

Answer 1

The acceleration of the particle is:

$f=at$

Now, acceleration is defined as the rate of change of velocity with respect to time.

So we can write acceleration as the derivative of velocity with respect to time.

$\displaystyle f = at \\ \\ \\ \implies \frac{dv}{dt}=at \\ \\ \\ \implies dv = at \, dt \\ \\ \\ \text{Integrating with proper limits} \\ \\ \text{Velocity is u at t=0.} \\ \\ \\ \implies \int\limits_{u}^{v}dv = \int\limits_{0}^{t} at \, dt \\ \\ \\ \implies \left[ v \right]_{u}^{v} = a \left[\frac{t^2}{2}\right]_0^t \\ \\ \\ \implies v-u = a \left[ \frac{t^2}{2} - 0\right] \\ \\ \\ \implies \boxed{\bold{v=u+\frac{at^2}{2}}}$

Thus, The Answer is Option (B)