The initial velocity of a particle is $u$ (at $t = 0$) and the acceleration $f$ is given by $at$. Which of the

Question

The initial velocity of a particle is $u$ (at $t = 0$) and the acceleration $f$ is given by $at$. Which of the following relations is valid?

Accepted Answer

1.  **Definition of Acceleration:** Acceleration $f$ is defined as the rate of change of velocity with respect to time: $f = \frac{dv}{dt}$ .
2.  **Given Condition:** The acceleration is given as a function of time: $f = at$.
3.  **Integration:** To find the velocity $v$ at time $t$, we integrate the acceleration function with respect to time:
    $\int_{u}^{v} dv = \int_{0}^{t} f \, dt = \int_{0}^{t} at \, dt$
    Here, the limits of integration for velocity are from initial velocity $u$ to final velocity $v$, and for time from $0$ to $t$.
4.  **Solving the Integral:**
    $v - u = a \left[ \frac{t^2}{2} \right]_{0}^{t}$
    $v - u = \frac{1}{2} at^2$
    $v = u + \frac{at^2}{2}$.
    *Note: The standard equation $v = u + at$  applies only to constant acceleration. Since acceleration here depends on time ($f \propto t$), calculus must be used.*

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