Question

If the displacement of a particle varies with time as√x = t + 7, then1) Velocity of the particle is inversely proportional to t2) Velocity of the particle is proportional to t^23) Velocity of the particle is proportional to√t4) The particle moves with constant acceleration

Answer 1

The relation between $x$ and $t$ is $$\begin{gathered} \sqrt{x}=t+7 \\ \Rightarrow x=(t+7{)}^2=t^2+14t+49 \end{gathered}$$
Now velocity, $v=\frac{dx}{dt}=2t+14$
and acceleration,
$a=\frac{dv}{dt}=2$ The object moves with a constant acceleration.