Question

A particle moves with a constant acceleration $a=2m{s}^{-2}$ along a straight line. If its moves with an initial velocity of $5m{s}^{-1}$, then obtain an expression for its instantaneous velocity.

Answer 1

We know that acceleration is the time rate of change of velocity, i.e., $a=dv/dt$ and differentiation is the inverse operation of integration acceleration, we can obtain the expression of velocity.
So, $v=\int adt=2\int dt=2t+c.....\left(i\right)$
where $c$ is the constant of integration and its value can be obtained from the initial conditions.
At $t=0,v=5m{s}^{-1}$. Putting these in Eq. (i), we have
$5=2\times 0+c$
$\Rightarrow c=5m{s}^{-1}$
Therefore, $v=2t+5$ is the required expression for the instantaneous velocity.